$f(E(G))=\\{f(x_{i}y_{j})=f(y_{j})-f(x_{i}):x_{i}y_{j}\in E(G)\\}=[1,q]$

Adding randomly $a_{i}$ new leaves $u_{i,j}\in L(x_{i})$ to each vertex
$x_{i}$ by adding new edges $x_{i}u_{i,j}$ for $j\in[1,a_{i}]$ and
$i\in[1,s]$, and adding randomly $b_{j}$ new leaves $v_{j,r}\in L(y_{j})$ to
each vertex $y_{j}$ by adding new edges $y_{j}v_{j,r}$ for $r\in[1,b_{j}]$ and
$j\in[1,t]$, it is allowed that some $a_{i}=0$ or some $b_{j}=0$. The
resultant graph is denoted as $G^{*}$. Let $A=\sum^{s}_{i=1}a_{i}$ and
$B=\sum^{t}_{j=1}b_{j}$, so $m=A+B$.

Step A-2. Define a graceful-difference total labeling $f_{gr}$ for $G$ by the
set-ordered graceful labeling $f$ in the following substeps:

Step A-2.1. $X$-dual transformation. Set

$f_{gr}(x_{i})=\max f(X)+\min f(X)-f(x_{i})=\max f(X)-f(x_{i}),~{}x_{i}\in X$

Step A-2.2. $Y$-dual transformation. Set

$f_{gr}(y_{j})=\max f(Y)+\min f(Y)-f(y_{j})=q+\min f(Y)-f(y_{j}),~{}y_{j}\in
Y$

Step A-2.3. Edge-dual transformation. Each edge $x_{i}y_{j}\in E(G)$ is
colored as

$f_{gr}(x_{i}y_{j})=\max f(E(G))+\min f(E(G))-f(x_{i}y_{j})=q+1-f(x_{i}y_{j})$

There is the following the graceful-difference constraint

${\begin{split}&\quad\big{|}|f_{gr}(x_{i})-f_{gr}(y_{j})|-f_{gr}(x_{i}y_{j})\big{|}=\big{|}f_{gr}(y_{j})-f_{gr}(x_{i})-f_{gr}(x_{i}y_{j})\big{|}\\\
&=\big{|}q+\min f(Y)-f(y_{j})-[\max
f(X)-f(x_{i})]-[q+1-f(x_{i}y_{j})]\big{|}\\\ &=\big{|}\min f(Y)-\max
f(X)+f(x_{i}y_{j})-f(y_{j})+f(x_{i})-1\big{|}\\\ &=\min f(Y)-\max
f(X)-1\end{split}}$ (54)

for each edge $x_{i}y_{j}\in E(G)$, which means that $f_{gr}$ is a graceful-
difference total labeling of $G$.

Step A-3. Parameterizing the graceful-difference total labeling. For integers
$k\geq 0$ and $d\geq 1$, we construct a $(k,d)$-constraint labeling $g_{gr}$
for $G$ as follows:

A-3.1. $g_{gr}(x_{i})=f_{gr}(x_{i})\cdot d$ for $x_{i}\in X$;

A-3.2. $g_{gr}(y_{j})=k+[f_{gr}(y_{j})-1]\cdot d$ for $y_{j}\in Y$; and

A-3.3. $g_{gr}(x_{i}y_{j})=k+[f_{gr}(x_{i}y_{j})-1]\cdot d$ for each edge
$x_{i}y_{j}\in E(G)$.

So, we obtain the vertex color set

$g_{gr}(V(G))=\\{f(x_{i})\cdot d:x_{i}\in X\\}\cup\\{k+[f_{gr}(y_{j})-1]\cdot
d:y_{j}\in Y\\}=g_{gr}(X)\cup g_{gr}(Y)$ (55)

and the edge color set

$g_{gr}(E(G))=\\{k+[f_{gr}(x_{i}y_{j})-1]\cdot d:x_{i}y_{j}\in
E(G)\\}=\\{k,k+d,\dots,k+(q-1)\cdot d\\}$ (56)

Moreover, for each edge $x_{i}y_{j}\in E(G)$, we use Eq.(54) to get the
graceful-difference constraint

${\begin{split}&\big{|}|g_{gr}(x_{i})-g_{gr}(y_{j})|-g_{gr}(x_{i}y_{j})\big{|}\\\
=&\big{|}k+[f_{gr}(y_{j})-1]\cdot d-f_{gr}(x_{i})\cdot
d-(k+[f_{gr}(x_{i}y_{j})-1]\cdot d)\big{|}\\\ =&[\min f(Y)-\max f(X)-1]\cdot
d\end{split}}$ (57)

Thereby, we claim that $g_{gr}$ is a graceful-difference $(k,d)$-total
labeling of $G$ by Definition 7.

Step A-4. Define a total coloring $h_{gr}$ for $G^{*}$ in the following
substeps:

Step A-4.1. Recolor the vertices and edges of $G$ as:

A-4.1. $h_{gr}(x_{i})=g_{gr}(x_{i})\cdot d$ for $x_{i}\in X$;

A-4.2. $h_{gr}(y_{j})=g_{gr}(y_{j})+m\cdot d$ for $y_{j}\in Y$; and

A-4.3. $h_{gr}(x_{i}y_{j})=g_{gr}(x_{i}y_{j})+m\cdot d$ for each edge
$x_{i}y_{j}\in E(G)$.

Then, there is the graceful-difference constraint

${\begin{split}\big{|}|h_{gr}(x_{i})-h_{gr}(y_{j})|-h_{gr}(x_{i}y_{j})\big{|}=&\big{|}|g_{gr}(y_{j})+m\cdot
d-g_{gr}(x_{i})|-g_{gr}(x_{i}y_{j})-m\cdot d\big{|}\\\ =&[\min f(Y)-\max
f(X)-1]\cdot d\end{split}}$ (58)

for each edge $x_{i}y_{j}\in E(G)$. Notice that the edge color set

$h_{gr}(E(G))=\\{k+m\cdot d,k+(m+1)\cdot d,\dots,k+(q+m-1)\cdot d\\}$

Step A-4.2. Color edges $y_{j}v_{j,r}$ for $v_{j,r}\in L(y_{j})$ with
$r\in[1,b_{j}]$ and $j\in[1,t]$.

Since each edge $y_{1}v_{1,r}$ generated by the leaves of $L(y_{1})$ is
colored with $h_{gr}(y_{1}v_{1,r})=k+(r-1)\cdot d$ for $r\in[1,b_{1}]$, and
$h_{gr}(y_{1}v_{1,r})=k+(b_{1}-1)\cdot d$; we have a formula

$h_{gr}(y_{j}v_{j,r})=k+(r-1)\cdot d+d\sum^{j-1}_{i=1}b_{i},~{}r\in[1,b_{j}]$
(59)

and

$h_{gr}(y_{j}v_{j,b_{j}})=k+(b_{j}-1)\cdot
d+d\sum^{j-1}_{i=1}b_{i}=k-d+d\sum^{j}_{i=1}b_{i}$

The last edge $y_{t}v_{t,b_{t}}$ is colored with

$h_{gr}(y_{t}v_{t,b_{t}})=k+(b_{t}-1)\cdot
d+d\sum^{t-1}_{i=1}b_{i}=k-d+d\sum^{t}_{i=1}b_{i}=k+(B-1)\cdot d$

Step A-4.3. Color edges $x_{i}u_{i,j}$ for $u_{i,j}\in L(x_{i})$ with
$j\in[1,a_{i}]$ and $i\in[1,s]$.

Since $h_{gr}(x_{s}u_{s,j})=k+(j-1)\cdot d+B\cdot d$ for $j\in[1,a_{s}]$,
$h_{gr}(x_{s}u_{s,a_{s}})=k+(a_{s}-1)\cdot d+B\cdot d$, we have a formula

$h_{gr}(x_{i}u_{i,j})=k+(j-1)\cdot d+B\cdot
d+d\sum^{s-i}_{r=1}a_{s-r+1},~{}j\in[1,a_{i}],~{}i\in[1,s-1]$ (60)

The last edge $x_{1}u_{1,a_{1}}$ is colored with

$h_{gr}(x_{1}u_{1,a_{1}})=k+(a_{1}-1)\cdot d+B\cdot
d+d\sum^{s-1}_{r=1}a_{s-r+1}=k+(A+B-1)\cdot d=k+(m-1)\cdot d$

Step A-5. Let $M_{gr}=[\min f(Y)-\max f(X)-1]$. Color each added leaf in the
following substeps:

Step A-5.1. Color each added leaf $u_{i,j}\in L(x_{i})$ for $j\in[1,a_{i}]$
and $i\in[1,s]$ with

$h_{gr}(u_{i,j})=h_{gr}(x_{i})+h_{gr}(x_{i}u_{i,j})+M_{gr}$

so we have the graceful-difference constraints

$\big{|}|h_{gr}(u_{i,j})-h_{gr}(x_{i})|-h_{gr}(x_{i}u_{i,j})\big{|}=M_{gr}\cdot
d,\quad x_{i}u_{i,j}\in E(G^{*})$

hold true.

Step A-5.2. Color each added leaf $v_{j,r}\in L(y_{j})$ with $r\in[1,b_{j}]$
and $j\in[1,t]$ with
$M_{gr}+h_{gr}(v_{j,r})=h_{gr}(y_{j})-h_{gr}(y_{j}v_{j,r})$, that are the
graceful-difference constraints

$\big{|}|h_{gr}(y_{j})-h_{gr}(v_{j,r})|-h_{gr}(y_{j}v_{j,r})\big{|}=M_{gr}\cdot
d,~{}y_{j}v_{j,r}\in E(G^{*})$

Step A-6. Return the graceful-difference $(k,d)$-total coloring $h_{gr}$ of
$G^{*}$.

The RLA-algorithm-A enables us to get the following result:

######  Theorem 23.

∗ Each tree admits a graceful-difference $(k,d)$-total coloring and an odd-
edge graceful-difference $(k,d)$-total coloring.

An example shown in Fig.28 is for illustrating the RLA-algorithm-A, and we can
see:

(a) A connected graph $G$ admits a set-ordered graceful-difference total
labeling $f_{gr}$ holding the graceful-difference constraint

$\big{|}|f_{gr}(x)-f_{gr}(y)|-f_{gr}(xy)\big{|}=0,~{}xy\in E(G)$

(b) a parameterized connected graph $G_{\textrm{p}}$ admitting a graceful-
difference $(k,d)$-total labeling $g_{gr}$ with the graceful-difference
constraint

$\big{|}|g_{gr}(x)-g_{gr}(y)|-g_{gr}(xy)\big{|}=0,~{}xy\in E(G_{\textrm{p}})$

(c) adding leaves to $G_{\textrm{p}}$ produces a connected graph
$G_{\textrm{p-leaf}}$;

(d) a connected graph $Gh_{\textrm{p-leaf}}$ admits a graceful-difference
$(k,d)$-total coloring $h_{gr}$ with the graceful-difference constraint

$\big{|}|h_{gr}(x)-h_{gr}(y)|-h_{gr}(xy)\big{|}=2d,~{}$

Figure 28:  Adding leaves to a connected graph $G$ makes a large scale
connected graph $Gh_{\textrm{p-leaf}}$ admitting a graceful-difference
$(k,d)$-total coloring.

######  Theorem 24.

∗ If a connected graph $G$ admits a set-ordered graceful labeling, then $G$
admits a graceful-difference $(k,d)$-total coloring and an odd-edge graceful-
difference $(k,d)$-total coloring.

######  Theorem 25.

(i) ∗ A connected graph $G$ admits a graceful-difference $(k,d)$-total
coloring if and only if there exists a tree $T$ admitting a graceful-
difference $(k,d)$-total coloring such that the result of vertex-coinciding
each group of vertices colored the same colors into one vertex is just $G$.

(ii) ∗ A connected bipartite graph $H$ admits an odd-edge graceful-difference
$(k,d)$-total coloring if and only if there exists a tree $T^{*}$ admitting an
odd-edge graceful-difference $(k,d)$-total coloring such that the result of
vertex-coinciding each group of vertices colored the same color in $T^{*}$
into one vertex is just $H$.

###  4.2 Parameterized edge-difference total labelings/colorings

∗ RLA-algorithm-B for the edge-difference $(k,d)$-total coloring.

Input: A connected bipartite $(p,q)$-graph $G$ admitting a set-ordered
graceful labeling $f$.

Output: A connected bipartite $(p+m,q+m)$-graph $G^{*}$ admitting an edge-
difference $(k,d)$-total coloring, where $G^{*}$ is the result of adding
randomly $m$ leaves to $G$.

Step B-1. Initialization. Since a connected bipartite $(p,q)$-graph $G$
admitting a set-ordered graceful labeling $f$, we have the vertex set
$V(G)=X\cup Y$ with $X\cap Y=\emptyset$, where

$X=\\{x_{1},x_{2},\dots,x_{s}\\},~{}Y=\\{y_{1},y_{2},\dots,y_{t}\\},~{}s+t=p=|V(G)|$

such that

$0=f(x_{1})<f(x_{2})<\cdots<f(x_{s})<f(y_{1})<f(y_{2})<\cdots<f(y_{t})=q$

with the edge color set

$f(E(G))=\\{f(x_{i}y_{j})=f(y_{j})-f(x_{i}):x_{i}y_{j}\in E(G)\\}=[1,q]$

Adding randomly $a_{i}$ new leaves $u_{i,j}\in L(x_{i})$ to each vertex
$x_{i}$ by adding new edges $x_{i}u_{i,j}$ for $j\in[1,a_{i}]$ and
$i\in[1,s]$, and adding randomly $b_{j}$ new leaves $v_{j,r}\in L(y_{j})$ to
each vertex $y_{j}$ by adding new edges $y_{j}v_{j,r}$ for $r\in[1,b_{j}]$ and
$j\in[1,t]$, it is allowed that some $a_{i}=0$ or some $b_{j}=0$. The
resultant graph is denoted as $G^{*}$. Let $A=\sum^{s}_{i=1}a_{i}$ and
$B=\sum^{t}_{j=1}b_{j}$, so $m=A+B$.

Step B-2. Define an edge-difference total labeling $f_{ed}$ for $G$ in the
following way:

B-2.1. $X$-dual transformation. Set

$f_{ed}(x_{i})=\max f(X)+\min f(X)-f(x_{i})=\max f(X)-f(x_{i}),~{}x_{i}\in X$

B-2.2. $Y$-dual transformation. Set

$f_{ed}(y_{j})=\max f(Y)+\min f(Y)-f(y_{j})=q+\min f(Y)-f(y_{j}),~{}y_{j}\in
Y$

B-2.3. $f_{ed}(x_{i}y_{j})=f(x_{i}y_{j})$ for each edge $x_{i}y_{j}\in E(G)$.
Since the edge-difference constraint

${\begin{split}f_{ed}(x_{i}y_{j})+|f_{ed}(x_{i})-f_{ed}(y_{j})|&=f_{ed}(x_{i}y_{j})+f_{ed}(y_{j})-f_{ed}(x_{i})\\\
&=f(x_{i}y_{j})+q+\min f(Y)-f(y_{j})-[\max f(X)-f(x_{i})]\\\
&=f(x_{i}y_{j})+q+\min f(Y)-\max f(X)-[f(y_{j})-f(x_{i})]\\\ &=q+\min
f(Y)-\max f(X),\quad x_{i}y_{j}\in E(G)\end{split}}$ (61)

hold true, so we claim that that $f_{ed}$ is a set-ordered edge-difference
total labeling of $G$.

Step B-3. Parameterizing the graceful-difference total labeling. For integers
$k\geq 0,d\geq 1$, we define a $(k,d)$-constraint labeling $g_{ed}$ for $G$ as
follows:

B-3.1. $g_{ed}(x_{i})=f_{ed}(x_{i})\cdot d$ for $x_{i}\in X$;

B-3.2. $g_{ed}(y_{j})=k+[f_{ed}(y_{j})-1]\cdot d$ for $y_{j}\in Y$;

B-3.3. $g_{ed}(x_{i}y_{j})=k+[f_{ed}(x_{i}y_{j})-1]\cdot d$ for each edge
$x_{i}y_{j}\in E(G)$.
So, the vertex color set

$g_{ed}(V(G))=\\{f(x_{i})\cdot d:x_{i}\in X\\}\cup\\{k+[f_{ed}(y_{j})-1]\cdot
d:y_{j}\in Y\\}=g_{ed}(X)\cup g_{ed}(Y)$

and the edge color set

$g_{ed}(E(G))=\\{k+[f_{ed}(x_{i}y_{j})-1]\cdot d:x_{i}y_{j}\in
E(G)\\}=\\{k,k+d,\dots,k+(q-1)\cdot d\\}$

Let $M_{ed}=q+\min f(Y)-\max f(X)$ shown in Eq.(61). We have that the edge-
difference constraint

${\begin{split}g_{ed}(x_{i}y_{j})+|g_{ed}(x_{i})-g_{ed}(y_{j})|=&k+[f_{ed}(x_{i}y_{j})-1]\cdot
d+|k+[f_{ed}(y_{j})-1]\cdot d-f_{ed}(x_{i})\cdot d|\\\
=&2(k-d)+f_{ed}(x_{i}y_{j})\cdot d+[f_{ed}(y_{j})-f_{ed}(x_{i})]\cdot d\\\
=&2(k-d)+M_{ed}\cdot d\end{split}}$ (62)

for each edge $x_{i}y_{j}\in E(G)$ holds true. Thereby, we claim that $g_{ed}$
is an edge-difference $(k,d)$-total labeling of $G$ according to Definition 7.

Step B-4. Define a total coloring $h_{ed}$ for $G^{*}$ in the following
substeps:

Step B-4.1. Recolor the vertices and edges of $G$ by setting

B-4.1. $h_{ed}(x_{i})=g_{ed}(x_{i})$ for $x_{i}\in X\subset V(G)$;

B-4.2. $h_{ed}(y_{j})=g_{ed}(y_{j})+m\cdot d$ for $y_{j}\in Y\subset V(G)$;
and

B-4.3. $h_{ed}(x_{i}y_{j})=g_{ed}(x_{i}y_{j})$ for each edge $x_{i}y_{j}\in
E(G)$.

Then, each edge $x_{i}y_{j}\in E(G)$ holds the edge-difference constraint

${\begin{split}h_{ed}(x_{i}y_{j})+|h_{ed}(y_{j})-h_{ed}(x_{i})|&=g_{ed}(x_{i}y_{j})+g_{ed}(y_{j})+m\cdot
d-g_{ed}(x_{i})\\\ &=2(k-d)+(M_{ed}+m)\cdot d\end{split}}$ (63)

true.

Step B-4.2. Color edges $x_{i}u_{i,j}$ for $u_{i,j}\in L(x_{i})$ with
$j\in[1,a_{i}]$ and $i\in[1,s]$. Since

$h_{ed}(x_{1}u_{1,j})=k+j\cdot d+(q-1)\cdot d,~{}j\in[1,a_{1}],\textrm{ and
}h_{ed}(x_{1}u_{1,a_{1}})=k+(q+a_{1}-1)\cdot d$

we have a formula

$h_{ed}(x_{i}u_{i,j})=k+j\cdot d+(q-1)\cdot
d+d\sum^{i-1}_{r=1}a_{r},~{}j\in[1,a_{i}],~{}i\in[1,s]$ (64)

The last edge $x_{s}u_{s,a_{s}}$ is colored with

$h_{ed}(x_{s}u_{s,a_{s}})=k+a_{s}\cdot d+(q-1)\cdot
d+d\sum^{s-1}_{r=1}a_{r}=k+(A+B-1)\cdot d=k+(q+A-1)\cdot d$

Step B-4.3. Color edges $y_{j}v_{j,r}$ for $v_{j,r}\in L(y_{j})$ with
$r\in[1,b_{j}]$ and $j\in[1,t]$. Because of $h_{ed}(y_{t}v_{t,r})=k+r\cdot
d+(q+A-1)\cdot d$ for $r\in[1,b_{t}]$, and $h_{ed}(y_{1}v_{1,r})=k+b_{t}\cdot
d+(q+A-1)\cdot d$. There is a formula

$h_{ed}(y_{j}v_{j,r})=k+r\cdot d+(q+A-1)\cdot
d+d\sum^{j-1}_{i=1}b_{t-i+1},~{}r\in[1,b_{j}],~{}j\in[1,t]$ (65)

as well as

$h_{ed}(y_{j}v_{j,b_{j}})=k+b_{j}\cdot d+(q+A-1)\cdot
d+d\sum^{j-1}_{i=1}b_{t-i+1}=k+(q+A-1)\cdot d+d\sum^{j}_{i=1}b_{t-i+1}$

The last edge $x_{s}u_{s,a_{s}}$ is colored with

$h_{ed}(y_{1}v_{1,b_{1}})=k+b_{1}\cdot d+(q+A-1)\cdot
d+d\sum^{t-1}_{i=1}b_{t-i+1}=k+(q+m-1)\cdot d$

Step B-4.4. Let $M^{*}=2(k-d)+(M_{ed}+m)\cdot d$ defined in Eq.(63). Color
added leaves $u_{i,j}\in L(x_{i})$ for $j\in[1,a_{i}]$ and $i\in[1,s]$ by

$h_{ed}(u_{i,j})=M^{*}-h_{ed}(x_{i}u_{i,j})+h_{ed}(x_{i}),~{}j\in[1,a_{i}],~{}i\in[1,s]$

Step B-4.5. Color added leaves $v_{j,r}\in L(y_{j})$ with $r\in[1,b_{j}]$ and
$j\in[1,t]$ as:

$h_{ed}(v_{j,r})=h_{ed}(y_{j})-[M^{*}-h_{ed}(y_{j}v_{j,r})],~{}r\in[1,b_{j}],~{}j\in[1,t]$

Also, we get the edge-difference constraint
$h_{ed}(y_{j}v_{j,r})+h_{ed}(y_{j})-h_{ed}(v_{j,r})=M^{*}$.

Step B-5. Return the edge-difference $(k,d)$-total coloring $h_{ed}$ of
$G^{*}$.

Fig.29 shows us examples for understanding the RLA-algorithm-B:

(a) A connected graph $T$ admits a set-ordered edge-difference total labeling
$f_{ed}$ holding the edge-difference constraint

$f_{ed}(xy)+|f_{ed}(x)-f_{ed}(y)|=10,~{}xy\in E(T)$

(b) a parameterized connected graph $T_{\textrm{p}}$ admitting a graceful-
difference $(k,d)$-total labeling $g_{ed}$ with the edge-difference constraint

$g_{ed}(xy)+|g_{ed}(x)-g_{ed}(y)|=2k+8d,~{}xy\in E(T_{\textrm{p}})$

(c) adding leaves to $T_{\textrm{p}}$ produces a connected graph
$T_{\textrm{p-leaf}}$;

(d) a connected graph $Th_{\textrm{p-leaf}}$ admits a graceful-difference
$(k,d)$-total coloring $h_{ed}$ with the edge-difference constraint

$h_{ed}(xy)+|h_{ed}(x)-h_{ed}(y)|=2k+22d,~{}$

Figure 29:  For understanding the RLA-algorithm-B for the edge-difference
$(k,d)$-total coloring.

The RLA-algorithm-B enables us to get the following result:

######  Theorem 26.

∗ Each tree admits an edge-difference $(k,d)$-total coloring and an odd-edge
edge-difference $(k,d)$-total coloring.

######  Theorem 27.

∗ If a connected graph $G$ admits a set-ordered graceful labeling, then $G$
admits an edge-difference $(k,d)$-total coloring and an odd-edge edge-
difference $(k,d)$-total coloring.

######  Theorem 28.

(i) ∗ A connected graph $G$ admits an edge-difference $(k,d)$-total coloring
if and only if there exists a tree $T$ admitting an edge-difference
$(k,d)$-total coloring such that the result of vertex-coinciding each group of
vertices colored the same colors into one vertex is just $G$.

(ii) ∗ A connected bipartite graph $H$ admits an odd-edge edge-difference
$(k,d)$-total coloring if and only if there exists a tree $T^{*}$ admitting an
odd-edge edge-difference $(k,d)$-total coloring such that the result of
vertex-coinciding each group of vertices colored the same color in $T^{*}$
into one vertex is just $H$.

###  4.3 Parameterized felicitous-difference total labelings/colorings

∗ RLA-algorithm-C for the felicitous-difference $(k,d)$-total coloring.

Input: A connected bipartite $(p,q)$-graph $G$ admitting a set-ordered
graceful labeling $f$.

Output: A connected bipartite $(p+m,q+m)$-graph $G^{*}$ admitting a
felicitous-difference $(k,d)$-total coloring, where $G^{*}$ is the result of
adding randomly $m$ leaves to $G$.

Step C-1. Initialization. Since a connected bipartite $(p,q)$-graph $G$
admitting a set-ordered graceful labeling $f$, we have the vertex set
$V(G)=X\cup Y$ with $X\cap Y=\emptyset$, where

$X=\\{x_{1},x_{2},\dots,x_{s}\\},~{}Y=\\{y_{1},y_{2},\dots,y_{t}\\},~{}s+t=p=|V(G)|$

such that

$0=f(x_{1})<f(x_{2})<\cdots<f(x_{s})<f(y_{1})<f(y_{2})<\cdots<f(y_{t})=q$

with the edge color set

$f(E(G))=\\{f(x_{i}y_{j})=f(y_{j})-f(x_{i}):x_{i}y_{j}\in E(G)\\}=[1,q]$

Adding randomly $a_{i}$ new leaves $u_{i,j}\in L(x_{i})$ to each vertex
$x_{i}$ by adding new edges $x_{i}u_{i,j}$ for $j\in[1,a_{i}]$ and
$i\in[1,s]$, and adding randomly $b_{j}$ new leaves $v_{j,r}\in L(y_{j})$ to
each vertex $y_{j}$ by adding new edges $y_{j}v_{j,r}$ for $r\in[1,b_{j}]$ and
$j\in[1,t]$, it is allowed that some $a_{i}=0$ or some $b_{j}=0$. The
resultant graph is denoted as $G^{*}$. Let $A=\sum^{s}_{i=1}a_{i}$ and
$B=\sum^{t}_{j=1}b_{j}$, so $m=A+B$.

Step C-2. Define a felicitous-difference total labeling $f_{fel}$ of $G$ in
the following way:

C-2.1. $X$-dual transformation. Each vertex $x_{i}\in X$ holds

$f_{fel}(x_{i})=\max f(X)+\min f(X)-f(x_{i})=\max f(X)-f(x_{i})$

C-2.2. $f_{fel}(y_{j})=f(y_{j})$ for $y_{j}\in Y$; and

C-2.3. $f_{fel}(x_{i}y_{j})=f(x_{i}y_{j})$ for each edge $x_{i}y_{j}\in E(G)$.
Because of the felicitous-difference constraint

${\begin{split}\big{|}f_{fel}(x_{i})+f_{fel}(y_{j})-f_{fel}(x_{i}y_{j})\big{|}&=\big{|}\max
f(X)-f(x_{i})+f(y_{j})-f(x_{i}y_{j})\big{|}=\max f(X)\end{split}}$ (66)

for each edge $x_{i}y_{j}\in E(G)$ holds true, so we claim that that $f_{fel}$
is a set-ordered felicitous-difference total labeling of $G$.

Step C-3. Parameterizing the felicitous-difference total labeling. For
integers $k\geq 0,d\geq 1$, we define a $(k,d)$-constraint labeling $g_{fel}$
for $G$ as follows:

C-3.1. $g_{fel}(x_{i})=f_{fel}(x_{i})\cdot d$ for $x_{i}\in X$;

C-3.2. $g_{fel}(y_{j})=k+[f_{fel}(y_{j})-1]\cdot d$ for $y_{j}\in Y$;

C-3.3. $g_{fel}(x_{i}y_{j})=k+[f_{fel}(x_{i}y_{j})-1]\cdot d$ for each edge
$x_{i}y_{j}\in E(G)$.
So, the vertex color set

$g_{fel}(V(G))=\\{f(x_{i})\cdot d:x_{i}\in
X\\}\cup\\{k+[f_{fel}(y_{j})-1]\cdot d:y_{j}\in Y\\}=g_{fel}(X)\cup
g_{fel}(Y)$

and the edge color set

$g_{fel}(E(G))=\\{k+[f_{fel}(x_{i}y_{j})-1]\cdot d:x_{i}y_{j}\in
E(G)\\}=\\{k,k+d,\dots,k+(q-1)\cdot d\\}$

By Eq.(71), the felicitous-difference constraint

${\begin{split}&\big{|}g_{fel}(x_{i})+g_{fel}(y_{j})-g_{fel}(x_{i}y_{j})\big{|}\\\
=&|f_{fel}(x_{i})\cdot d+k+[f_{fel}(y_{j})-1]\cdot
d-(k+[f_{fel}(x_{i}y_{j})-1]\cdot d)|\\\ =&f_{fel}(x_{i})\cdot
d+f_{fel}(y_{j})\cdot d-f_{fel}(x_{i}y_{j})\cdot d\\\ =&\max f(X)\cdot
d\end{split}}$ (67)

for each edge $x_{i}y_{j}\in E(G)$ holds true. Thereby, we claim that
$g_{fel}$ is a felicitous-difference $(k,d)$-total labeling of $G$ from
Definition 7.

Step C-4. Define a total coloring $h_{fel}$ for $G^{*}$ in the following
substeps:

Step C-4.1. Recolor the vertices and edges of $G$ by setting

C-4.1. $h_{fel}(x_{i})=g_{fel}(x_{i})$ for $x_{i}\in X\subset V(G)$;

C-4.2. $h_{fel}(y_{j})=g_{fel}(y_{j})+m\cdot d$ for $y_{j}\in Y\subset V(G)$;
and

C-4.3. $h_{fel}(x_{i}y_{j})=g_{fel}(x_{i}y_{j})$ for each edge $x_{i}y_{j}\in
E(G)$.

So, there are the following felicitous-difference constraints

${\begin{split}|h_{fel}(y_{j})+h_{fel}(x_{i})-h_{fel}(x_{i}y_{j})|&=g_{fel}(x_{i})+g_{fel}(y_{j})+m\cdot
d-g_{fel}(x_{i}y_{j})\\\ &=[\max f(X)+m]\cdot d,~{}x_{i}y_{j}\in
E(G)\end{split}}$ (68)

Step C-4.2. Color edges $x_{i}u_{i,j}$ for $u_{i,j}\in L(x_{i})$ with
$j\in[1,a_{i}]$ and $i\in[1,s]$.

From $h_{fel}(x_{1}u_{1,j})=k+j\cdot d+(q-1)\cdot d$ for $j\in[1,a_{1}]$,
$h_{fel}(x_{1}u_{1,a_{1}})=k+(q+a_{1}-1)\cdot d$, we get a formula

$h_{fel}(x_{i}u_{i,j})=k+j\cdot d+(q-1)\cdot
d+d\sum^{i-1}_{r=1}a_{r},~{}j\in[1,a_{i}],~{}i\in[1,s]$ (69)

The last edge $x_{s}u_{s,a_{s}}$ is colored with

$h_{fel}(x_{s}u_{s,a_{s}})=k+a_{s}\cdot d+(q-1)\cdot
d+d\sum^{s-1}_{r=1}a_{r}=k+(A+B-1)\cdot d=k+(q+A-1)\cdot d$

Step C-4.3. Color edges $y_{j}v_{j,r}$ for $v_{j,r}\in L(y_{j})$ with
$r\in[1,b_{j}]$ and $j\in[1,t]$.

Since

$h_{fel}(y_{t}v_{t,r})=k+r\cdot d+(q+A-1)\cdot d,~{}r\in[1,b_{t}],\textrm{ and
}h_{fel}(y_{1}v_{1,r})=k+b_{t}\cdot d+(q+A-1)\cdot d$

we have a formula

$h_{fel}(y_{j}v_{j,r})=k+r\cdot d+(q+A-1)\cdot
d+d\sum^{j-1}_{i=1}b_{t-i+1},~{}r\in[1,b_{j}]$ (70)

and

$h_{fel}(y_{j}v_{j,b_{j}})=k+b_{j}\cdot d+(q+A-1)\cdot
d+d\sum^{j-1}_{i=1}b_{t-i+1}=k+(q+A-1)\cdot d+d\sum^{j}_{i=1}b_{t-i+1}$

The last edge $y_{1}v_{1,b_{1}}$ is colored with

$h_{fel}(y_{1}v_{1,b_{1}})=k+b_{1}\cdot d+(q+A-1)\cdot
d+d\sum^{t-1}_{i=1}b_{t-i+1}=k+(q+m-1)\cdot d$

Step C-4.4. Let $M\,^{\prime}=[\max f(X)+m]\cdot d$ defined in Eq.(68). Color
added leaves $u_{i,j}\in L(x_{i})$ for $j\in[1,a_{i}]$ and $i\in[1,s]$ as:

$h_{fel}(u_{i,j})=M\,^{\prime}+h_{fel}(x_{i}u_{i,j})-h_{fel}(x_{i}),~{}j\in[1,a_{i}],~{}i\in[1,s]$

which is just the felicitous-difference constraint.

Step C-4.5. Color the added leaves $v_{j,r}\in L(y_{j})$ with $r\in[1,b_{j}]$
and $j\in[1,t]$ as:

$h_{fel}(v_{j,r})=M\,^{\prime}+h_{fel}(y_{j}v_{j,r})-h_{fel}(y_{j}),~{}r\in[1,b_{j}],~{}j\in[1,t]$

Also, the felicitous-difference constraint

$h_{fel}(v_{j,r})+h_{fel}(y_{j})-h_{fel}(y_{j}v_{j,r})=M\,^{\prime}$

holds true for each edge $y_{j}v_{j,r}\in E(G^{*})$.

Summarizing all steps together, we confirm that $h_{fel}$ is really a
felicitous-difference $(k,d)$-total coloring of $G^{*}$.

Step C-5. Return the felicitous-difference $(k,d)$-total coloring $h_{fel}$ of
$G^{*}$.

Fig.30 is for understanding the RLA-algorithm-C, we can see:

(a) A connected graph $L$ admits a set-ordered felicitous-difference total
labeling $f_{fel}$ holding the felicitous-difference constraint

$\big{|}f_{fel}(x)+f_{fel}(y)-f_{fel}(xy)\big{|}=3,~{}xy\in E(L)$

(b) a parameterized connected graph $L_{\textrm{p}}$ admitting a felicitous-
difference $(k,d)$-total labeling $g_{fel}$ with the felicitous-difference
constraint

$\big{|}g_{fel}(x)+g_{fel}(y)-g_{fel}(xy)\big{|}=3d,~{}xy\in
E(L_{\textrm{p}})$

(c) adding leaves to $L_{\textrm{p}}$ produces a connected graph
$L_{\textrm{p-leaf}}$;

(d) a connected graph $Lh_{\textrm{p-leaf}}$ admits a felicitous-difference
$(k,d)$-total coloring $h_{fel}$ with the felicitous-difference constraint

$\big{|}h_{fel}(x)+h_{fel}(y)-h_{fel}(xy)\big{|}=17d,~{}xy\in
E(Lh_{\textrm{p-leaf}})$

Figure 30:  For understanding the RLA-algorithm-C for the felicitous-
difference $(k,d)$-total coloring.

The RLA-algorithm-C enables us to get the following result:

######  Theorem 29.

∗ Each tree admits a felicitous-difference $(k,d)$-total coloring and an odd-
edge felicitous-difference $(k,d)$-total coloring.

######  Theorem 30.

∗ If a connected graph $G$ admits a set-ordered graceful labeling, then $G$
admits a felicitous-difference $(k,d)$-total coloring and an odd-edge
felicitous-difference $(k,d)$-total coloring.

######  Theorem 31.

(i) ∗ A connected graph $G$ admits a felicitous-difference $(k,d)$-total
coloring if and only if there exists a tree $T$ admitting a felicitous-
difference $(k,d)$-total coloring such that such that the result of vertex-
coinciding each group of vertices colored the same colors into one vertex is
just $G$.

(ii) ∗ A connected bipartite graph $H$ admits an odd-edge felicitous-
difference $(k,d)$-total coloring if and only if there exists a tree $T^{*}$
admitting an odd-edge felicitous-difference $(k,d)$-total coloring such that
such that the result of vertex-coinciding each group of vertices colored the
same color in $T^{*}$ into one vertex is just $H$.

###  4.4 Parameterized edge-magic total labelings/colorings

∗ RLA-algorithm-D for the edge-magic $(k,d)$-total coloring.

Input: A connected bipartite $(p,q)$-graph $G$ admitting a set-ordered
graceful labeling $f$.

Output: A connected bipartite $(p+m,q+m)$-graph $G^{*}$ admitting an edge-
magic $(k,d)$-total coloring, where $G^{*}$ is the result of adding randomly
$m$ leaves to $G$.

Step D-1. Initialization. Since a connected bipartite $(p,q)$-graph $G$
admitting a set-ordered graceful labeling $f$, we have the vertex set
$V(G)=X\cup Y$ with $X\cap Y=\emptyset$, where

$X=\\{x_{1},x_{2},\dots,x_{s}\\},~{}Y=\\{y_{1},y_{2},\dots,y_{t}\\},~{}+t=p=|V(G)|$

such that

$0=f(x_{1})<f(x_{2})<\cdots<f(x_{s})<f(y_{1})<f(y_{2})<\cdots<f(y_{t})=q$

with the edge color set

$f(E(G))=\\{f(x_{i}y_{j})=f(y_{j})-f(x_{i}):x_{i}y_{j}\in E(G)\\}=[1,q]$

Adding randomly $a_{i}$ new leaves $u_{i,j}\in L(x_{i})$ to each vertex
$x_{i}$ by adding new edges $x_{i}u_{i,j}$ for $j\in[1,a_{i}]$ and
$i\in[1,s]$, and adding randomly $b_{j}$ new leaves $v_{j,r}\in L(y_{j})$ to
each vertex $y_{j}$ by adding new edges $y_{j}v_{j,r}$ for $r\in[1,b_{j}]$ and
$j\in[1,t]$, it is allowed that some $a_{i}=0$ or some $b_{j}=0$. The
resultant graph is denoted as $G^{*}$. Let $A=\sum^{s}_{i=1}a_{i}$ and
$B=\sum^{t}_{j=1}b_{j}$, so $m=A+B$.

Step D-2. Define an edge-magic total labeling $f_{mag}$ for $G$ in the
following way:

D-2.1. $X$-dual transformation. Each vertex $x_{i}\in X$ holds

$f_{mag}(x_{i})=\max f(X)+\min f(X)-f(x_{i})=\max f(X)-f(x_{i})$

D-2.2. $f_{mag}(y_{j})=f(y_{j})$ for $y_{j}\in Y$; and

D-2.3. Edge-dual transformation. Each edge $x_{i}y_{j}\in E(G)$ holds

$f_{mag}(x_{i}y_{j})=\max f(E(G))+\min f(E(G))-f(x_{i}y_{j})$

Since each edge $x_{i}y_{j}$ of $G$ holds the edge-magic constraint

${\begin{split}&\quad f_{mag}(x_{i})+f_{mag}(x_{i}y_{j})+f_{mag}(y_{j})\\\
&=\max f(X)-f(x_{i})+\max f(E(G))+\min f(E(G))-f(x_{i}y_{j})+f(y_{j})\\\
&=\max f(E(G))+\min f(E(G))+\max f(X)\\\ &=\max f(X)+q+1\end{split}}$ (71)

true, so $\lambda_{mag}$ is an edge-magic total labeling of $G$.

Step D-3. Parameterizing the felicitous-difference total labeling. For
integers $k\geq 0,d\geq 1$, we define a $(k,d)$-constraint labeling $g_{mag}$
for $G$ as follows:

D-3.1. $g_{mag}(x_{i})=f_{mag}(x_{i})\cdot d$ for $x_{i}\in X$;

D-3.2. $g_{mag}(y_{j})=k+[f_{mag}(y_{j})-1]\cdot d$ for $y_{j}\in Y$;

D-3.3. $g_{mag}(x_{i}y_{j})=k+[f_{mag}(x_{i}y_{j})-1]\cdot d$ for each edge
$x_{i}y_{j}\in E(G)$.
So, the vertex color set

$g_{mag}(V(G))=\\{f(x_{i})\cdot d:x_{i}\in
X\\}\cup\\{k+[f_{mag}(y_{j})-1]\cdot d:y_{j}\in Y\\}=g_{mag}(X)\cup
g_{mag}(Y)$

and the edge color set

$g_{mag}(E(G))=\\{k+[f_{mag}(x_{i}y_{j})-1]\cdot d:x_{i}y_{j}\in
E(G)\\}=\\{k,k+d,\dots,k+(q-1)\cdot d\\}$

By Eq.(71), we have that the edge-magic constraint

${\begin{split}&g_{mag}(x_{i})+g_{mag}(x_{i}y_{j})+g_{mag}(y_{j})\\\
=&f_{mag}(x_{i})\cdot d+(k+[f_{mag}(x_{i}y_{j})-1]\cdot
d)+(k+[f_{mag}(y_{j})-1]\cdot d)\\\
=&2(k-d)+[f_{mag}(x_{i})+f_{mag}(x_{i}y_{j})+f_{mag}(y_{j})]\\\ =&2k+[\max
f(X)+q-1]\cdot d\end{split}}$ (72)

for each edge $x_{i}y_{j}\in E(G)$ holds true. Thereby, we claim that
$g_{mag}$ is an edge-magic $(k,d)$-total labeling of $G$ by Definition 7.

Step D-4. Define a total coloring $h_{mag}$ for $G^{*}$ in the following
substeps:

Step D-4.1. Recolor the vertices and edges of $G$ by setting

D-4.1. $h_{mag}(x_{i})=g_{mag}(x_{i})$ for $x_{i}\in X\subset V(G)$;

D-4.2. $h_{mag}(y_{j})=g_{mag}(y_{j})+m\cdot d$ for $y_{j}\in Y\subset V(G)$;
and

D-4.3. $h_{mag}(x_{i}y_{j})=g_{mag}(x_{i}y_{j})+m\cdot d$ for each edge
$x_{i}y_{j}\in E(G)$.

Thereby, we have that the edge-magic constraint

${\begin{split}h_{mag}(y_{j})+h_{mag}(x_{i}y_{j})+h_{mag}(x_{i})&=g_{mag}(x_{i})+g_{mag}(x_{i}y_{j})+m\cdot
d+g_{mag}(y_{j})+m\cdot d\\\ &=2k+[\max f(X)+q+2m-1]\cdot d\end{split}}$ (73)

for each edge $x_{i}y_{j}\in E(G)$ holds true.

Step D-4.2. Color edges $y_{j}v_{j,r}$ for $v_{j,r}\in L(y_{j})$ with
$r\in[1,b_{j}]$ and $j\in[1,t]$.

Since $h_{mag}(y_{t}v_{t,r})=k+(r-1)\cdot d$ for $r\in[1,b_{t}]$ and
$h_{mag}(y_{t}v_{t,b_{t}})=k+(b_{t}-1)\cdot d$, we have a formula

$h_{mag}(y_{j}v_{j,r})=k+(r-1)\cdot
d+d\sum^{j-1}_{i=1}b_{t-i+1},~{}r\in[1,b_{j}],j\in[2,t]$ (74)

and

$h_{mag}(y_{j}v_{j,b_{j}})=k+(b_{j}-1)\cdot
d+d\sum^{j-1}_{i=1}b_{t-i+1}=k+d\sum^{j}_{i=1}b_{t-i+1}$

The last edge $y_{1}v_{1,b_{1}}$ is colored with

$h_{mag}(y_{1}v_{1,b_{1}})=k+(b_{1}-1)\cdot
d+d\sum^{t-1}_{i=1}b_{t-i+1}=k+(B-1)\cdot d$

Step D-4.3. Color edges $x_{i}u_{i,j}$ for $u_{i,j}\in L(x_{i})$ with
$j\in[1,a_{i}]$ and $i\in[1,s]$.

From $h_{mag}(x_{s}u_{s,j})=k+j\cdot d+(B-1)\cdot d$ for $j\in[1,a_{1}]$, and

$h_{mag}(x_{s}u_{s,a_{s}})=k+a_{s}\cdot d+(B-1)\cdot d$

we get a formula

$h_{mag}(x_{i}u_{i,j})=k+j\cdot d+(B-1)\cdot
d+d\sum^{s-i}_{r=1}a_{s-r+1},~{}j\in[1,a_{i}],~{}i\in[2,s]$ (75)

The last edge $x_{1}u_{1,a_{1}}$ is colored with

$h_{mag}(x_{1}u_{1,a_{1}})=k+a_{1}\cdot d+(B-1)\cdot
d+d\sum^{s-1}_{r=1}a_{s-r+1}=k+(A+B-1)\cdot d=k+(m-1)\cdot d$

Step D-4.4. Let $M\,^{\prime\prime}=2k+[\max f(X)+q+2m-1]\cdot d$ defined in
Eq.(73). Color added leaves $u_{i,j}\in L(x_{i})$ for $j\in[1,a_{i}]$ and
$i\in[1,s]$ as:

$h_{mag}(u_{i,j})=M\,^{\prime\prime}-h_{mag}(x_{i}u_{i,j})-h_{mag}(x_{i}),~{}j\in[1,a_{i}],~{}i\in[1,s]$

which is just the edge-magic constraint.

Step D-4.5. Color added leaves $v_{j,r}\in L(y_{j})$ with $r\in[1,b_{j}]$ and
$j\in[1,t]$ as:

$h_{mag}(v_{j,r})=M\,^{\prime\prime}-h_{mag}(y_{j}v_{j,r})-h_{mag}(y_{j}),~{}r\in[1,b_{j}],~{}j\in[1,t]$

which is just the edge-magic constraint
$h_{mag}(v_{j,r})+h_{mag}(y_{j}v_{j,r})+h_{mag}(y_{j})=M\,^{\prime\prime}$.

Thereby, we claim that $h_{mag}$ is really an edge-magic $(k,d)$-total
coloring of $G^{*}$.

Step D-5. Return the edge-magic $(k,d)$-total coloring $h_{mag}$ of $G^{*}$.

Fig.31 is for understanding the RLA-algorithm-D for the edge-magic
$(k,d)$-total coloring, there are:

(a) A connected graph $H$ admits a set-ordered edge-magic total labeling
$f_{mag}$ holding the edge-magic constraint

$f_{mag}(x)+f_{mag}(xy)+f_{mag}(y)=3,~{}xy\in E(L)$

(b) a parameterized connected graph $H_{\textrm{p}}$ admitting an edge-magic
$(k,d)$-total labeling $g_{mag}$ with the edge-magic constraint

$g_{mag}(x)+g_{mag}(xy)+g_{mag}(y)=2k+11d,~{}xy\in E(H_{\textrm{p}})$

(c) adding leaves to $H_{\textrm{p}}$ produces a connected graph
$H_{\textrm{p-leaf}}$;

(d) a connected graph $Hh_{\textrm{p-leaf}}$ admits an edge-magic
$(k,d)$-total coloring $h_{mag}$ with the edge-magic constraint

$h_{mag}(x)+h_{mag}(xy)+h_{mag}(y)=2k+35d,~{}xy\in E(Hh_{\textrm{p-leaf}})$

Figure 31:  For understanding the RLA-algorithm-D for the edge-magic
$(k,d)$-total coloring.
Figure 32:  Adding leaf two times, edge-difference $(k,d)$-total coloring.
Figure 33:  Adding leaf two times, graceful-difference $(k,d)$-total coloring.
Figure 34:  Adding leaf two times, felicitous-difference $(k,d)$-total
coloring.

The RLA-algorithm-D enables us to get the following result:

######  Theorem 32.

∗ Each tree admits an edge-magic $(k,d)$-total coloring and an odd-edge edge-
magic $(k,d)$-total coloring.

######  Theorem 33.

∗ If a connected graph $G$ admits a set-ordered graceful labeling, then $G$
admits an edge-magic $(k,d)$-total coloring and an odd-edge edge-magic
$(k,d)$-total coloring.

######  Theorem 34.

(i) ∗ A connected graph $G$ admits an edge-magic $(k,d)$-total coloring if and
only if there exists a tree $T$ admitting an edge-magic $(k,d)$-total coloring
such that such that the result of vertex-coinciding each group of vertices
colored the same colors into one vertex is just $G$.

(ii) ∗ A connected bipartite graph $H$ admits an odd-edge edge-magic
$(k,d)$-total coloring if and only if there exists a tree $T^{*}$ admitting an
odd-edge edge-magic $(k,d)$-total coloring such that such that the result of
vertex-coinciding each group of vertices colored the same color in $T^{*}$
into one vertex is just $H$.

##  5 Parameterized labelings on trees having perfect matchings

###  5.1 Basic labelings on trees having perfect matchings

######  Definition 18.

[18, 19] Let $T$ be a $(p,q)$-tree having a perfect matching $M$. Suppose that
each leaf of $T$ is an end of some matching edge of $M$, and $M(L)$ is the set
of all leaves of $T$, and furthermore the graph $T-M(L)$ obtained by deleting
all leaves of $T$ is just a tree having $k=|V(T-M(L))|$ vertices.

(1) [18] Suppose that $T$ admits an odd-elegant labeling $f_{1}$. If
$|f_{1}(u)-f_{1}(v)|=1$ for each matching edge $uv\in M$ holds true, and the
elements of $f_{1}(E(M))$ form an arithmetic progression having $|M|$ terms
with the first term $1$ and the common difference 4. Then we call $f_{1}$
_arithmetic matching odd-elegant labeling_ (_arm-odd-elegant labeling_) of
$T$, and $T$ is an _arm-odd-elegant graph_.

(2) [18] Suppose that $T$ admits a super felicitous labeling $f_{2}$. If
$|f_{2}(u)-f_{2}(v)|=k$ for every matching edge $uv\in M$, we call $f_{2}$
_arithmetic matching super felicitous labeling_ (_arm-super felicitous
labeling_) of $T$, and $T$ an _arm-super felicitous graph_.

(3) [18] Suppose that $T$ admits a super edge-magic graceful labeling $f_{3}$.
If $|f_{3}(u)-f_{3}(v)|=k$ for each matching edge $uv\in M$, and the elements
of $f_{3}(E(T))$ produce an arithmetic progression having $|M|$ terms with the
first term $p+1$ and the common difference $2$. Then we call $f_{3}$
_arithmetic matching super edge-magic graceful labeling_ (_arm-super edge-
magic graceful labeling_) of $T$, and $T$ an _arm-super edge-magic graceful
graph_.

(4) [19] Suppose that $T$ admits a super edge-magic total labeling $f_{4}$. If
$|f_{4}(u)-f_{4}(v)|=k$ for each matching edge $uv\in M$, and the elements of
$f_{4}(E(T))$ yield an arithmetic progression having $|M|$ terms with the
first term $2p-1$ and the common difference $-2$. Then we call $f_{4}$
_arithmetic matching super edge-magic total labeling_ (_arm-super edge-magic
total labeling_) of $T$, and $T$ an _arm-super edge-magic total graph_.
$\square$

######  Definition 19.

[19] Let $T$ be a tree with a perfect matching $M$.

(1) Suppose that $T$ admits a super $(k,d)$-edge antimagic labeling $g$
defined in Definition 5. If $|g(u)-g(v)|=k$ for each matching edge $uv\in M$,
and

(i) the elements of $g(E(T))$ produce an arithmetic progression having $|M|$
terms with the first term $p+1$ and the common difference $2$;

(ii) $g(u)+g(v)=b_{i}$ forms a sequence $\\{b_{i}\\}$ being an arithmetic
progression having $|M|$ terms with the first term $k+2$ and the common
difference $2$, where $n=|M|$.

Then we call $g$ an _arithmetic matching super edge antimagic $(k,d)$-total
labeling_ (_arm-super edge-antimagic $(k,d)$-total labeling_) of $T$, and $T$
an _arm-super edge-antimagic $(k,d)$-total labeling graph_.

(2) Suppose that $T$ admits a $(k,d)$-arithmetic labeling $h$ defined in
Definition 5. If $|h(u)-h(v)|=k$ for each matching edge $uv\in M$, and the
elements of $h(E(T))$ produce an arithmetic progression having $|M|$ terms
with the first term $k$ and the common difference $2d$. Then we call $h$ an
_arithmetic matching $(k,d)$-arithmetic labeling_ (_arm- $(k,d)$-arithmetic
labeling_) of $T$, and $T$ an _arm- $(k,d)$-arithmetic graph_. $\square$

Figure 35: For illustrating Definition 18 and Definition 19: (a) An arm-odd-
elegant labeling; (b) an arm-super felicitous labeling; (c) an arm-super edge-
magic graceful labeling; (d) an arm-super $(|X|+p+3,2)$-edge antimagic total
labeling; (e) an arm-super edge-total graceful labeling; (f) an
arm-$(k,d)$-arithmetic labeling.

###  5.2 $(k,d)$-constraint labelings on trees with perfect matchings

######  Definition 20.

[20] Let $T$ be a $(p,q)$-tree having a perfect matching $M$, and let

$S_{q-1,k,d}=\\{k,k+d,\dots,k+(q-1)d\\}$,

$S\,^{\prime}_{k,d}=\\{k,k+2d,\dots,k+2(q-1)d\\}$,

$S\,^{\prime\prime}_{k,d}=\\{k,k+d,\dots,k+(p+q-1)d\\}$
for integers $k\geq 1$ and $d\geq 1$. Suppose that each leaf of $T$ is an end
of some matching edge of $M$, and $M(L)$ is the set of all leaves of $T$, and
furthermore the graph $T-M(L)$ obtained by deleting all leaves of $T$ is a
tree having $c=|V(T-M(L))|$ vertices. The abbreviation of “arithmetic
matching” is “arm”.

  1. S-1.

An _arm- $(k,d)$-totally odd (resp. even) graceful labeling_ $f$ of $G$ holds
$f(V(G))\subseteq[0,k+2(q-1)d]$, $f(x)\neq f(y)$ for distinct vertices $x,y\in
V(G)$ and

$f(E(G))=\\{|f(u)-f(v)|:\ uv\in E(G)\\}=S\,^{\prime}_{k,d}$

Meanwhile, $f(u)+f(v)=k+2(q-1)d$ for each matching edge $uv\in M$.

  2. S-2.

An _arm- $(k,d)$-felicitous labeling_ $f$ of $G$ holds
$f(V(G))\subseteq[0,k+(q-1)d]$, $f(x)\neq f(y)$ for distinct vertices $x,y\in
V(G)$ and

$f(E(G))=\\{f(uv)=f(u)+f(v)~{}(\bmod~{}qd):uv\in E(G)\\}=S_{q-1,k,d}$

and furthermore, $f$ is _super_ if

$f(V(G))=\left[0,\left(\frac{p}{2}-1\right)d\right]\bigcup\left[k+\frac{pd}{2},k+(p-1)d\right]$

Meanwhile, $|f(u)-f(v)|=k+cd$ for each matching edge $uv\in M$.

  3. S-3.

An _arm-edge-magic $(k,d)$-total labeling_ $f$ of $G$ holds $f(V(G)\cup
E(G))\subseteq[0,k+(p+q-1)d]$ such that $f(u)+f(v)+f(uv)=a$ for any edge
$uv\in E(G)$, where the magic constant $a$ is a fixed integer; and
furthermore, $f$ is _super_ if $f(V(G))=[1,pd/2]\cup[k+pd/2,k+qd]$. Meanwhile,
$|f(u)-f(v)|=k+(c-1)d$ for each matching edge $uv\in M$ and the elements of
the edge color set $f(E(T))$ yield an arithmetic progression having $|M|$
terms with the first term $k+2(p-1)d$ and the common difference $-2d$.

  4. S-4.

An _arm-edge-magic $(k,d)$-graceful labeling_ $f$ of $G$ holds $f(V(G)\cup
E(G))\subseteq[0,k+(p+q-1)d]$ such that $\mid f(u)+f(v)-f(uv)\mid=a$ for any
edge $uv\in E(G)$, where the magic constant $a$ is a fixed integer; and
furthermore, $f$ is _super_ if $f(V(G))=[1,pd/2]\cup[k+pd/2,k+qd]$. Meanwhile,
$|f(u)-f(v)|=k+(c-1)d$ for each matching edge $uv\in M$ and the elements of
the edge color set $f(E(T))$ yield an arithmetic progression having $|M|$
terms with the first term $k+pd$ and the common difference $2d$.

  5. S-5.

Suppose that $T$ admits a $(k,d)$-arithmetic labeling $f$. If $|f(u)-f(v)|=k$
for each matching edge $uv\in M$, and the elements of the edge color set
$f(E(T))$ produce an arithmetic progression having $|M|$ terms with the first
term $k$ and the common difference $2d$. Then we call $f$ _arm-
$(k,d)$-arithmetic labeling_ of $T$.

  6. S-6.

An _arm- $(k,d)$-totally odd (resp. even)-elegant labeling_ $f$ of $G$ holds
$f(V(G))\subset[0,k+2(q-1)d]$, $f(u)\neq f(v)$ for distinct vertices $u,v\in
V(G)$, and

$f(E(G))=\\{f(uv)=f(u)+f(v)~{}(\bmod~{}k+2qd):uv\in
E(G)\\}=S\,^{\prime}_{k,d}$

Meanwhile, $|f(u)-f(v)|=k$ for each matching edge $uv\in M$ and the elements
of the edge color set $f(E(T))$ yield an arithmetic progression having $|M|$
terms with the first term $k$ and the common difference $4d$. $\square$

Figure 36: For illustrating Definition 20: (a) An arm-$(k,d)$-totally odd
(resp. even) graceful labeling; (b) an arm-$(k,d)$-super felicitous labeling;
(c) an arm-super edge-magic $(k,d)$-total labeling; (d) an arm-$(k,d)$-super
edge-magic graceful labeling.

######  Definition 21.

[17] Suppose that $T$ is a tree of $p$ vertices and $q$ edges, and has a
perfect matching $M$ and its vertex bipartition $V(T)=X\cup Y$ with $X\cap
Y=\emptyset$ such that each edge $xy\in E(T)$ holds $x\in X$ and $y\in Y$. Let
$S_{m,\alpha,k\beta}=\\{\alpha,\alpha+k\beta,\dots,\alpha+k(m-1)\beta\\}$ with
integers $k\geq 1$, $m\geq 2$, $\alpha\geq 0$ and $\beta\geq 1$, and let $c$
be a constant, as well as $\lambda=\lfloor\frac{p}{2}\rfloor$. There are the
following restrictive conditions:

  1. C-1.

$\sigma:V(T)\rightarrow S_{p-1,0,\beta}\cup S_{2p-3,\alpha,\beta}$, and
$\sigma(x)\neq\sigma(w)$ for any pair of vertices $x,w\in V(T)$;

  2. C-2.

$\sigma:V(T)\rightarrow S_{\lambda,0,\beta}\cup S_{p,\alpha,\beta}$, and
$\sigma(x)\neq\sigma(w)$ for any pair of vertices $x,w\in V(T)$;

  3. C-3.

$\sigma(E(T))=\\{\sigma(xy)=\sigma(x)+\sigma(y):xy\in
E(T)\\}=S_{q,\alpha,2\beta}$;

  4. C-4.

$\sigma(E(T))=\\{\sigma(xy)=\sigma(x)+\sigma(y):xy\in
E(T)\\}=S_{q,\alpha,\beta}$;

  5. C-5.

Induced edge label $\sigma(xy)=|\sigma(x)-\sigma(y)|$ for each edge $xy\in
E(T)$, and $\sigma(E(T))=\\{\sigma(xy):xy\in E(T)\\}=S_{q,\alpha,\beta}$;

  6. C-6.

Induced edge label $\sigma(xy)=|\sigma(x)-\sigma(y)|$ for each edge $xy\in
E(T)$, and $\sigma(E(T))=\\{\sigma(xy):xy\in E(T)\\}=S_{q,\alpha,2\beta}$;

  7. C-7.

Induced edge label
$\sigma(xy)-\alpha=\sigma(x)+\sigma(y)-\alpha~{}(\bmod~{}q\beta)$ for each
edge $xy\in E(T)$, and $\sigma(E(T))=\\{\sigma(xy):xy\in
E(T)\\}=S_{q,\alpha,\beta}$;

  8. C-8.

$\sigma:V(T)\cup E(T)\rightarrow S_{\lambda+1,0,\beta}\cup
S_{p+q,\alpha,\beta}$, and $\sigma(x)\neq\sigma(w)$ for any pair of vertices
$x,w\in V(T)$;

  9. C-9.

$\sigma(X)=S_{\lambda,0,\beta}$ and $\sigma(Y)=S_{p-1,\alpha,\beta}\setminus
S_{\lambda-1,\alpha,\beta}$;

  10. C-10.

$\sigma(X)=S_{\lambda,0,\beta}$ and $\sigma(Y)=S_{p,\alpha,\beta}\setminus
S_{\lambda,\alpha,\beta}$;

  11. C-11.

$\sigma(X)=S_{\lambda,0,\beta}\setminus\\{0\\}$ and
$\sigma(Y)=S_{p,\alpha,\beta}\setminus S_{\lambda,\alpha,\beta}$;

———— _matching-set_

  12. C-12.

$|\sigma(x)-\sigma(y)|=\alpha+\frac{1}{2}p\beta$ for each matching edge $xy\in
M$;

  13. C-13.

$|\sigma(x)-\sigma(y)|=\alpha+\frac{1}{2}(p-2)\beta$ for each matching edge
$xy\in M$;

  14. C-14.

$\sigma(x)+\sigma(y)=\alpha+2(q-1)\beta$ for each matching edge $xy\in M$;

  15. C-15.

$|\sigma(x)-\sigma(y)|=\alpha$ for each matching edge $xy\in M$;

  16. C-16.

$|\sigma(x)-\sigma(y)|=\alpha+(q-1)\beta$ for each matching edge $xy\in M$;

  17. C-17.

$\sigma(M)=S_{|M|,\alpha,2\beta}$;

  18. C-18.

$\sigma(M)=S_{|M|,\alpha,4\beta}$;

  19. C-19.

$\sigma(M)=\\{\alpha+2(p-1)\beta,\alpha+2(p-2)\beta,\dots,\alpha+2(p-|M|)\beta\\}$;

  20. C-20.

$\sigma(M)=\\{\alpha+p\beta,\alpha+(2+p)\beta,\dots,\alpha+[2(|M|-1)+p]\beta\\}$;

———— _edge-magic_

  21. C-21.

$\sigma(x)+\sigma(y)+\sigma(xy)=c$ for each edge $xy\in E(T)$;

  22. C-22.

$\sigma(xy)+|\sigma(x)-\sigma(y)|=c$ for each edge $xy\in E(T)$;

  23. C-23.

$|\sigma(x)+\sigma(y)-\sigma(xy)|=c$ for each edge $xy\in E(T)$;

  24. C-24.

$\big{|}\sigma(xy)-|\sigma(x)-\sigma(y)|\big{|}=c$ for each edge $xy\in E(T)$.

Then the labeling $\sigma$ is:

  1. Label-1.

[6] an _$(\alpha,\beta)$ -graceful labeling_ if C-2, C-5 and C-16 hold true;

  2. Label-2.

a _set-ordered $(\alpha,\beta)$-graceful labeling_ if C-2, C-5, C-16 and C-9
hold true (see an example shown in Fig.37 (a));

  3. Label-3.

an _$(\alpha,\beta)$ totally odd/even-graceful labeling_ if C-1, C-6 and C-14
hold true (see an example shown in Fig.37 (b));

  4. Label-4.

an _arm- $(\alpha,\beta)$ totally odd/even-elegant labeling_ if C-1, C-3, C-15
and C-18 hold true (see an example shown in Fig.37 (c));

  5. Label-5.

an _arm- $(\alpha,\beta)$ felicitous labeling_ if C-2, C-7 and C-12 hold true;

  6. Label-6.

a _super arm- $(\alpha,\beta)$ felicitous labeling_ if C-2, C-7, C-12 and C-10
hold true (see an example shown in Fig.38 (e));

  7. Label-7.

an _arm- $(\alpha,\beta)$ arithmetic labeling_ if C-8, C-15, C-4 and C-17 hold
true (see an example shown in Fig.38 (d));

———— _$W$ -magic_

  8. Label-8.

an _arm- $(\alpha,\beta)$ edge-magic total labeling_ if C-8, C-21, C-13 and
C-19 hold true;

  9. Label-9.

a _super arm- $(\alpha,\beta)$ edge-magic total labeling_ if C-8, C-21, C-13,
C-11 and C-19 hold true (see an example shown in Fig.38 (f));

  10. Label-10.

an _arm- $(\alpha,\beta)$ edge-magic graceful labeling_ if C-8, C-13, C-23 and
C-20 hold true;

  11. Label-11.

a _super arm- $(\alpha,\beta)$ edge-magic graceful labeling_ if C-8, C-13,
C-23, C-11 and C-20 hold true (see an example shown in Fig.39 (g));

  12. Label-12.

an _arm- $(\alpha,\beta)$ edge-difference labeling_ if C-1, C-22 and C-16 hold
true;

  13. Label-13.

a _super arm- $(\alpha,\beta)$ edge-difference labeling_ if C-1, C-22, C-16
and C-9 hold true (see an example shown in Fig.39 (h));

  14. Label-14.

an _arm- $(\alpha,\beta)$ graceful-difference labeling_ if C-1, C-24 and C-16
hold true;

  15. Label-15.

a _super arm- $(\alpha,\beta)$ graceful-difference labeling_ if C-1, C-24,
C-16 and C-9 hold true (see an example shown in Fig.39 (i)). $\square$

Figure 37: $p=44$, $q=43$ and $|M|=22$ in Definition 21: (a) A strongly set-
ordered $(\alpha,\beta)$-graceful labeling; (b) an arm-$(\alpha,\beta)$
totally odd/even-graceful labeling; (c) an arm-$(\alpha,\beta)$ totally
odd/even-elegant labeling, cited from [17]. Figure 38: $p=44$, $q=43$ and
$|M|=22$ in Definition 21: (d) An arm-$(\alpha,\beta)$ arithmetic labeling;
(e) a super arm-$(\alpha,\beta)$ felicitous labeling; (f) a super
arm-$(\alpha,\beta)$ edge-magic total labeling, cited from [17]. Figure 39:
$p=44$, $q=43$ and $|M|=22$ in Definition 21: (g) a super arm-$(\alpha,\beta)$
edge-magic graceful labeling; (h) a super arm-$(\alpha,\beta)$ graceful-
difference labeling; (i) a super arm-$(\alpha,\beta)$ graceful-difference
labeling, cited from [17].

######  Definition 22.

[17] Let $f$ be a strongly set-ordered $(\alpha,\beta)$-graceful labeling of a
tree $T$, we have the following equivalent transformations:

(1) An arm-$(\alpha,\beta)$ graceful-difference labeling $\sigma$ is defined
as: $\sigma(w)=f(w)$ for $w\in V(T)$, $\sigma(xy)=2\alpha+(q-1)\beta-f(xy)$
for each edge $xy\in E(T)$.

(2) An arm-$(\alpha,\beta)$ edge-difference labeling $\sigma$ is defined as:
$\sigma(x)=\max f(X)+\min f(X)-f(x)$ for $x\in X$, $\sigma(y)=\max f(Y)+\min
f(Y)-f(y)$ for $y\in Y$, $\sigma(xy)=\max f(E(T))+\min f(E(T))-f(xy)$ for each
edge $xy\in E(T)$. $\square$

######  Lemma 35.

[17] A _leaf-matching tree_ $T$ with a perfect matching $M$ has its leaf set
$L(T)$ such that each matching edge $uv\in M$ has one end $u$ in $L(T)$. If
the leaf-matching tree $T$ admits a strongly set-ordered graceful labeling,
then it admits the following pairwise equivalent labelings:

(1) strongly set-ordered graceful labeling;

(2) strongly $(\alpha,\beta)$-graceful labeling;

(3) arm-$(\alpha,\beta)$-totally odd/even-graceful labeling;

(4) arm-$(\alpha,\beta)$-totally odd/even-elegant labeling;

(5) arm-$(\alpha,\beta)$ arithmetic labeling;

(6) arm-$(\alpha,\beta)$ super felicitous labeling;

(7) arm-$(\alpha,\beta)$ super edge-magic total labeling;

(8) arm-$(\alpha,\beta)$ super edge-magic graceful labeling.

######  Lemma 36.

Suppose that a tree $T$ has a perfect matching $M(T)$ and admits a strongly
graceful labeling $\theta$. There are three operations:

$(i)$ Exchanging-edge operation. A new strongly graceful labeling $\varphi$ of
$T$ holding $\varphi(x)=\theta(y)$ and $\varphi(y)=\theta(x)$ for each
matching edge $xy\in M(T)$.

$(ii)$ Decreasing-leaf operation. If a matching edge $xy\in M(T)$ satisfies
the neighborhood $N(x)=\\{y,x_{i}:i\in[1,d_{x}-1]\\}$ with
$d_{x}=\textrm{deg}_{T}(x)\geq 3$ and $\textrm{deg}_{T}(y)=1$. Deleting the
edges $xx_{j}$ and joins $y$ with $x_{j}$ for $j\in[s+1,d_{x}-1]$ with $0\leq
s<d_{x}-1$ produces a new tree $H$ such that $H$ is also strongly graceful,
and its perfect matching $M(H)$ keeps

$|L(T)\cap V(M(T))|-1=|L(H)\cap V(M(H))|$

$(iii)$ Increasing-leaf operation. If a matching edge $xy\in M(T)$ satisfies
$\textrm{deg}_{T}(x)\geq 2$ and $\textrm{deg}_{T}(y)\geq 2$. A process of (a)
moving the edge $xy$ from $T$; (b) identifying the vertex $x$ with the vertex
$y$ into one denoted as $x_{0}$; and (c) adding a new vertex $y_{0}$ to join
with $x_{0}$ will build up another tree $T\,^{\prime}$ having a strongly
graceful labeling. Furthermore, the perfect matching
$M(T\,^{\prime})=\\{x_{0}y_{0}\\}\cup\big{(}M(T)\setminus\\{xy\\}\big{)}$
holds

$|L(T)\cap V(M(T))|+1=|L(T\,^{\prime})\cap V(M(T\,^{\prime}))|$

###### Proof.

In order to prove the operation $(i)$, let $\theta$ be a strongly graceful
labeling of a tree $T$ with $n$ vertices and a perfect matching $M(T)$. We
define a labeling $\varphi$ for $T$ as: $\varphi(u)=\theta(v)$ and
$\varphi(v)=\theta(u)$ for each edge $uv\in M(T)$. Clearly,
$\varphi(u)+\varphi(v)=\theta(v)+\theta(u)=n-1$ and
$|\varphi(u)-\varphi(v)|=|\theta(v)-\theta(u)|$ for each edge $uv\in M(T)$.
For each edge $xy\in E(T)\setminus M(T)$, there are two edges
$xx\,^{\prime},yy\,^{\prime}\in M(T)$ so that

${\begin{split}|\varphi(x)-\varphi(y)|&=|\theta(x\,^{\prime})-\theta(y\,^{\prime})|=\big{|}(n-1)-\theta(y\,^{\prime})-\big{[}(n-1)-\theta(x\,^{\prime})\big{]}\big{|}=|\theta(y)-\theta(x)|\end{split}}$

We obtain the vertex color set $\theta(V(T))=\varphi(V(T))$ and the edge color
set $\theta(E(T))=\varphi(E(T))$, as well as $\varphi(u)+\varphi(v)=n-1$ for
each edge $uv\in M(T)$. Therefore, $\varphi$ is really a strongly graceful
labeling of $T$.

To prove the operation $(ii)$, we assume that there is an edge $uv\in M(T)$
such that the neighborhood $N(u)=\\{v,u_{i}:i\in[1,d_{u}-1]\\}$ and $N(u)\cap
L(T)=\\{v\\}$ since $N(T)$ is a perfect matching, where
$d_{u}=\textrm{deg}_{T}(u)\geq 3$. We have a tree $H$ defined by the
hypothesis of the operation $(ii)$. Since $H-uv$ has two components
$T\,^{\prime}$ and $T\,^{\prime\prime}$, without loss of generality, the
vertex $u$ lies in $T\,^{\prime}$. Notice that $M(H)=M(T)$. We define a
labeling $\varphi$ for $H$ as: $\varphi(x)=\theta(x)$ with $x\in
V(T\,^{\prime})$; $\varphi(v)=\theta(v)$; $\varphi(x)=\theta(y)$ and
$\varphi(y)=\theta(x)$ for each edge $xy\in E(T\,^{\prime\prime})\cap M(T)$.

Since $V(H)=V(T)$, so two vertex color sets $\varphi(V(H))=\theta(V(T))$. For
edges $xy\in E(T\,^{\prime})$, we have

$|\varphi(x)-\varphi(y)|=|\theta(x)-\theta(y)|,~{}|\varphi(u)-\varphi(v)|=|\theta(u)-\theta(v)|$

and every edge $xy\in E(T\,^{\prime\prime})\cap M(T)$ holds
$|\varphi(x)-\varphi(y)|=|\theta(x)-\theta(y)|$ true.

Considering each edge $xy\in E(T\,^{\prime\prime})\setminus M(T)$, so there
are two edges $xx\,^{\prime},yy\,^{\prime}\in E(T\,^{\prime\prime})\cap M(T)$.

Notice that $\varphi(x)=\theta(x\,^{\prime})$ and
$\varphi(y)=\theta(y\,^{\prime})$, and $n=|V(H)|$ $(=|V(T)|)$. We have

$|\varphi(x)-\varphi(y)|=|\theta(x\,^{\prime})-\theta(y\,^{\prime})|=|(n-1)-\theta(y\,^{\prime})-[(n-1)-\theta(x\,^{\prime})]|=|\theta(y)-\theta(x)|$

since $\theta(x)+\theta(x\,^{\prime})=n-1$ and
$\theta(y)+\theta(y\,^{\prime})=n-1$ according to the definition of a strongly
graceful labeling. Therefore, $\varphi(E(H))=\theta(E(T))$, which means that
$\varphi$ is a strongly graceful labeling of $H$. Clearly,

$|L(T)\cap V(M(T))|-1=|L(H)\cap V(M(H))|$

Notice that in the above proof of the operation $(ii)$, the strongly graceful
tree $H$ and a strongly graceful labeling $\varphi$ of $H$ enable us to obtain
the original tree $T$ and a strongly graceful labeling $\theta$ of $T$.
Thereby, the inverse of the proof of the operation $(ii)$ is just the proof of
the operation $(iii)$. ∎

######  Definition 23.

[17] Let a $(p,q)$-graph $G$ have a perfect matching $M$ and a strongly
labeling $\theta$. If there are two edges $uv\in E(G)$ and $xy\notin E(G)$
holding $\theta(xy)=\theta(uv)$, then the operation of removing the edge $uv$
from $G$ and adding the edge $xy$ to the remainder $G-uv$ is called an _edge-
mismatched transfer-operation_ , the resultant graph is denoted as $G-uv+xy$,
called _$\pm$ -edge graph_, and it admits a strongly labeling $\pi$ induced by
the strongly labeling $\theta$.

By the operation defined in Definition 23 and the operations stated in Lemma
36, Sun _et al._ , in the article [17], presented a conjecture as follows:

######  Conjecture 1.

[17] Any tree with a perfect matching can be transformed into some path with a
perfect matching by the edge-mismatched transfer-operation defined in
Definition 23 such that they both admit strongly graceful labelings.

Figure 40: A scheme for illustrating Definition 23 and Conjecture 1.

###  5.3 The $\pm e$-operations and the $\pm e$-graphs

For the needs of theoretical research and practical application, we propose
the following problems:

######  Definition 24.

∗ Let $T$ be a uncolored tree of $q$ edges. If each uncolored $\pm$-edge graph
$T_{i+1}=T_{i}-u_{i}v_{i}+x_{i}y_{i}$ is just a tree of $q$ edges holding
$u_{i}v_{i}\in E(T_{i})$ and $x_{i}y_{i}\notin E(T_{i})$, where $T_{1}=T$,
then we put all uncolored $\pm$-edge trees
$T_{i+1}=T_{i}-u_{i}v_{i}+x_{i}y_{i}$ into a set $S_{\pm e}(T)$, and write

$T_{i}\rightarrow^{1}_{\pm e}T_{i+1}$ (76)

called _$\pm e$ -operation graph homomorphism_. The set $S_{\pm e}(T)$ is
called _$\pm e$ -tree set_, and $T$ is called _source_ of the $\pm e$-tree set
$S_{\pm e}(T)$. $\square$

######  Problem 9.

About Definition 24, we have the following facts:

(1) The $\pm e$-tree set $S_{\pm e}(T)$ contains a unique path of $q$ edges;

(2) There is a tree $H\in S_{\pm e}(T)$ holding diameters $D(H)\leq D(G)$ for
each tree $G\in S_{\pm e}(T)$, then $H$ is a star $K_{1,q}$.
Consider the following questions:

  1. (i)

If $T$ admits a $W$-constraint coloring/labeling, does each tree $G\in S_{\pm
e}(T)$ admit a $W$-constraint coloring/labeling too? If it is not so, find the
subset $S_{W\textrm{-c-c}}(T)\subset S_{\pm e}(T)$, such that each tree in the
subset $S_{W\textrm{-c-c}}(T)$ admits a $W$-constraint coloring/labeling.

  2. (ii)

For two different trees $T,T\,^{\prime}$ of $q$ edges, are there $\pm e$-tree
sets $S_{\pm e}(T)=S_{\pm e}(T\,^{\prime})$?

  3. (iii)

If $T$ is a tree with a perfect matching as $q+1=2m$, determine the subset
$S_{match}(T)\subseteq S_{\pm e}(T)$ such that each tree of $S_{match}(T)$ has
a perfect matching. Moreover, if the perfect matching tree $T$ admits a
strongly $W$-constraint coloring/labeling, does each tree in $S_{match}(T)$
admit a strongly $W$-constraint coloring/labeling too?

  4. (iv)

Determine all trees with diameter $D$ in $S_{\pm e}(T)$ for each diameter
$D\in[3,q-1]$, and the subset $S_{dia}(T;D)\subset S_{\pm e}(T)$ collects
these trees with diameter $D$.

  5. (v)

Determine all trees with maximal degree $\Delta$ in $S_{\pm e}(T)$ for each
maximal degree $\Delta\in[3,q-1]$, and we use a subset
$S_{max}(T;\Delta)\subset S_{\pm e}(T)$ to collect these trees with maximal
degree $\Delta$.

  6. (vi)

Determine all trees with $m$ leaves in $S_{\pm e}(T)$ for each integer
$m\in[3,q-1]$, and we put them into the subset $S_{leaf}(T;m)\subset S_{\pm
e}(T)$.

  7. (vii)

Find three integers $D_{0},\Delta_{0}$ and $m_{0}$ falling into $3\leq
D_{0},\Delta_{0},m_{0}\leq q-1$, such that

$S_{\pm e}(T)=S_{dia}(T;D_{0})\cup S_{max}(T;\Delta_{0})\cup
S_{leaf}(T;m_{0})$ (77)

holds true. Moreover, consider Eq.(77) for one of the following situations:

(1) $D_{0},\Delta_{0}$ and $m_{0}$ are prime numbers.

(2) $D_{0},\Delta_{0}$ and $m_{0}$ are odd integers.

(3) $D_{0},\Delta_{0}$ and $m_{0}$ are even integers.

(4) Each of odd integers $D_{0},\Delta_{0}$ and $m_{0}$ is just a product of
two prime numbers.

(5) Each of even integers $D_{0},\Delta_{0}$ and $m_{0}$ is just a sum of two
prime numbers.

  8. (viii)

Find subsets $S_{para}(T;D,\Delta,m)\subset S_{\pm e}(T)$ for each group of
integers $D,\Delta,m$ holding $3\leq D,\Delta,m\leq q-1$, such that each tree
$G\in S_{para}(T;D,\Delta,m)$ just has $m$ leaves, maximal degree
$\Delta(G)=\Delta$ and diameter $D(G)=D$.

  9. (ix)

For two trees $G\,^{\prime}$ and $G\,^{\prime\prime}$ of $S_{\pm e}(T)$, if
there are $T_{i+1}=T_{i}-u_{i}v_{i}+x_{i}y_{i}\in S_{\pm e}(T)$ for
$i\in[1,r]$, such that $G\,^{\prime}=T_{1}$ and $G\,^{\prime\prime}=T_{r+1}$,
then by Eq.(76), we get a series of _$\pm e$ -operation graph homomorphisms_

$G\,^{\prime}=T_{1}\rightarrow^{1}_{\pm e}T_{2}\rightarrow^{1}_{\pm
e}T_{3}\rightarrow^{1}_{\pm e}~{}\cdots~{}\rightarrow^{1}_{\pm
e}T_{r}\rightarrow^{1}_{\pm e}T_{r+1}=G\,^{\prime\prime}$ (78)

and we write it in a short term $G\,^{\prime}\rightarrow^{[1,r]}_{\pm
e}G\,^{\prime\prime}$. If $r$ is the smallest in all $\pm e$-operation graph
homomorphisms $G\,^{\prime}\rightarrow^{[1,t]}_{\pm e}G\,^{\prime\prime}$, we
write $r=D^{\pm e}_{homo}\langle G\,^{\prime},G\,^{\prime\prime}\rangle$,
called _$\pm e$ -operation graph homomorphism distance_. Determine the $\pm
e$-tree set $S_{\pm e}(T)$’s diameter

$D(S_{\pm e}(T))=\max\\{D^{\pm e}_{homo}\langle
G\,^{\prime},G\,^{\prime\prime}\rangle:~{}G\,^{\prime},G\,^{\prime\prime}\in
S_{\pm e}(T)\\}$ (79)

  10. (x)

Suppose that a connected graph $J$ admits a vertex graph-coloring
$\theta:V(J)\rightarrow U^{*}$, where $U^{*}$ is one of the $\pm e$-tree sets
$S_{match}(T)$, $S_{max}(T;\Delta)$, $S_{dia}(T;D)$, $S_{leaf}(T;m)$, and
$S_{\pm e}(T)$, such that $\theta(u)=\theta(v)-x_{v}y_{v}+w_{u}z_{u}$ with
$x_{v}y_{v}\in E(\theta(v))$ and $w_{u}z_{u}\notin E(\theta(v))$ for each edge
$uv\in E(H)$, and moreover there is an edge $u\,^{\prime}v\,^{\prime}\in E(H)$
holding $\theta(u\,^{\prime})=T_{i}$ and $\theta(v\,^{\prime})=T_{i+1}$ if
there is $T_{i+1}=T_{i}-u_{i}v_{i}+x_{i}y_{i}$ for two trees $T_{i},T_{i+1}\in
U^{*}$. We call $J$ the _$\pm e$ -tree-intersected graph_ of the $\pm e$-tree
set $U^{*}$, like that in _hypergraphs_ (Ref. [35]). Immediately, we get two
diameters $D(J)$ and $D(S_{\pm e}(T))$ holding $D(J)=D(S_{\pm e}(T))$ true.

Since the $\pm e$-tree-intersected graph $J$ describes indirectly the
topological structure of each $\pm e$-tree set $U^{*}\in\\{S_{match}(T)$,
$S_{max}(T;\Delta)$, $S_{dia}(T;D)$, $S_{leaf}(T;m)$, $S_{\pm e}(T)\\}$, so it
is important to characterize the topological properties of the $\pm e$-tree-
intersected graph $J$, such as, Hamilton property, Euler’s property, diameter
$D(J)$, maximal degree $\Delta(J)$, clique, girth, cycles, various
connectivity, colorings, and so on.

######  Definition 25.

Suppose that a tree $T^{c}$ of $q$ edges admits a vertex labeling
$g:V(T)\rightarrow[0,q]$ holding the vertex color set $f(V(T))=[0,q]$, then we
get a tree set $S_{\pm e}(T^{c})$, called _colored $\pm e$-tree set_, by the
_$\pm e$ -operation_ $T^{c}_{i+1}=T^{c}_{i}-u_{i}v_{i}+x_{i}y_{i}$ holding
$u_{i}v_{i}\in E(T^{c}_{i})$ and $x_{i}y_{i}\notin E(T^{c}_{i})$, where
$T^{c}_{1}=T^{c}$, and two colored trees $T^{c}_{i+1}$ and $T^{c}_{i}$ form a
_$\pm e$ -operation colored-graph homomorphism_

$T^{c}_{i}\rightarrow^{1}_{\pm e}T^{c}_{i+1}$ (80)

such that any pair of trees $H\,^{\prime}$ and $H\,^{\prime\prime}$ of $S_{\pm
e}(T^{c})$ corresponds to $T^{c}_{i+1}=T^{c}_{i}-u_{i}v_{i}+x_{i}y_{i}\in
S_{\pm e}(T^{c})$ for $i\in[1,A]$ with $H\,^{\prime}=T^{c}_{1}$ and
$H\,^{\prime\prime}=T^{c}_{A+1}$, so we have a series of _$\pm e$ -operation
graph homomorphisms_

$H\,^{\prime}=T^{c}_{1}\rightarrow^{1}_{\pm e}T^{c}_{2}\rightarrow^{1}_{\pm
e}T^{c}_{3}\rightarrow^{1}_{\pm e}~{}\cdots~{}\rightarrow^{1}_{\pm
e}T^{c}_{A}\rightarrow^{1}_{\pm e}T^{c}_{A+1}=H\,^{\prime\prime}$ (81)

also, $H\,^{\prime}\rightarrow^{[1,A]}_{\pm e}H\,^{\prime\prime}$, and the
$\pm e$-operation graph homomorphism distance between two trees $H\,^{\prime}$
and $H\,^{\prime\prime}$ is denoted as

$D^{\pm e}_{homo}\langle
H\,^{\prime},H\,^{\prime\prime}\rangle=\min\left\\{A:~{}H\,^{\prime}\rightarrow^{[1,A]}_{\pm
e}H\,^{\prime\prime}\right\\}$

over all series of $\pm e$-operation graph homomorphisms shown in Eq.(81). The
parameter

$D(S_{\pm e}(T^{c}))=\max\\{D^{\pm e}_{homo}\langle
H\,^{\prime},H\,^{\prime\prime}\rangle:~{}H\,^{\prime},H\,^{\prime\prime}\in
S_{\pm e}(T^{c})\\}$ (82)

is called the _diameter_ of the $\pm e$-tree set $S_{\pm e}(T^{c})$. $\square$

######  Problem 10.

Recall Cayley’s formula $\tau(K_{n})=n^{n-2}$ (Ref. [2]), which tells us that
there are $n^{n-2}$ different colored spanning trees in a complete graph
$K_{n}$ admitting a vertex labeling $f:V(K_{n})\rightarrow[1,n]$ such that the
vertex color set $f(V(K_{n}))=[1,n]$, so the colored $\pm e$-tree set $S_{\pm
e}(T^{c})$ defined in Definition 25 holds $|S_{\pm e}(T^{c})|=q^{q-2}$ true.

About Definition 25, we propose the following problems:

  1. C-1\.

Determine the subset $S_{match}(T^{c})$ of the colored $\pm e$-tree set
$S_{\pm e}(T^{c})$ if the colored tree $T^{c}$ is a tree with a perfect
matching.

  2. C-2\.

If the vertex labeling $g$ is just a graceful labeling of $T^{c}$, find out
the subset $S_{grace}(T^{c})$ of the colored $\pm e$-tree set $S_{\pm
e}(T^{c})$, such that each tree of $S_{grace}(T^{c})$ admits a graceful
labeling too.

  3. C-3\.

If the vertex labeling $g$ is just a set-ordered graceful labeling of $T^{c}$,
find out the subset $S^{order}_{grace}(T^{c})$ of the colored $\pm e$-tree set
$S_{\pm e}(T^{c})$, such that each tree of $S^{order}_{grace}(T^{c})$ admits a
set-ordered graceful labeling too.

  4. C-4\.

Determine the subset $S^{match}_{grace}(T^{c})$ of the colored $\pm e$-tree
set $S_{\pm e}(T^{c})$ if the colored tree $T^{c}$ has a perfect matching and
the vertex labeling $g$ is just a strongly graceful labeling of $T^{c}$, such
that each tree of $S^{match}_{grace}(T^{c})$ has a perfect matching and admits
a strongly graceful labeling as $q+1=2m$. Clearly, for any perfect matching
tree $G\in S^{match}_{grace}(T^{c})$, we have a sequence
$\\{T^{c}_{i}-u_{i}v_{i}+x_{i}y_{i}\\}^{r}_{i=1}$, such that
$G=T^{c}_{r}-u_{r}v_{r}+x_{r}y_{r}$, also, we have a _$\pm e$ -operation
colored-graph homomorphism_

$T^{c}\rightarrow^{[1,r]}_{\pm e}G$ (83)

through the _$\pm e$ -operation colored-graph homomorphisms_
$T^{c}_{i}\rightarrow^{1}_{\pm e}T^{c}_{i+1}$ (see Eq.(80)) with each
$T^{c}_{i}$ is a perfect matching tree for $i\in[1,r]$. Here, the integer $r$
is the smallest $\pm e$-operation colored-graph homomorphism distance
$r=D^{\pm e}_{homo}\langle T^{c},G\rangle$ over all $\pm e$-operation colored-
graph homomorphisms $T^{c}\rightarrow^{[1,t]}_{\pm e}G$. Find a perfect
matching tree $G^{*}\in S^{match}_{grace}(T^{c})$ such that the $\pm
e$-operation colored-graph homomorphism distance $D^{\pm e}_{homo}\langle
T^{c},G^{*}\rangle$ is largest in the subset $S^{match}_{grace}(T^{c})$.

  5. C-5\.

A connected graph $H$ admits a vertex graph-coloring $F:V(H)\rightarrow
S^{*}$, where $S^{*}$ is one of subsets $S^{match}_{grace}(T^{c})$,
$S_{match}(T^{c})$, $S^{order}_{grace}(T^{c})$, and $S_{\pm e}(T^{c})$ defined
above, such that $F(u)=F(v)-x_{v}y_{v}+w_{u}z_{u}$ with $x_{v}y_{v}\in
E(F(v))$ and $w_{u}z_{u}\notin E(F(v))$ for each edge $uv\in E(H)$, and
moreover there is an edge $u\,^{\prime}v\,^{\prime}\in E(H)$ holding
$F(u\,^{\prime})=T^{c}_{i}$ and $F(v\,^{\prime})=T^{c}_{i+1}$ if there is
$T^{c}_{i+1}=T^{c}_{i}-u_{i}v_{i}+x_{i}y_{i}$ for two trees
$T^{c}_{i},T^{c}_{i+1}\in S^{*}$. We call $H$ the _$\pm e$ -tree-intersected
graph_ of the colored $\pm e$-tree set $S^{*}$.

Because of the $\pm e$-tree-intersected graph $H$ describes indirectly the
topological structure of each $\pm e$-tree set
$S^{*}\in\\{S^{match}_{grace}(T^{c})$, $S_{match}(T^{c})$,
$S^{order}_{grace}(T^{c})$, $S_{\pm e}(T^{c})\\}$, then the topological
structures of each $\pm e$-tree set $S^{*}$ can be described by the
topological properties of each $\pm e$-tree-intersected graph $H$, such as,
Hamilton property, Euler’s property, diameter $D(H)$, maximal degree
$\Delta(H)$, clique, colorings, girth, cycles, various connectivity, _etc_
(Ref. [35]). Notice that each element $F(w)$ of the Topcode-matrix
$T_{code}(H,F)$ is a colored $\pm e$-tree corresponding to a Topcode-matrix,
so the Topcode-matrix $T_{code}(H,F)$ is like that appeared in _Tensor_.

######  Definition 26.

∗ For a uncolored $(q+1,q)$-tree $G$, the $\pm e$-graph set $P_{\pm e}(G)$
contains all $\pm$-edge graphs $G_{i+1}=G_{i}-u_{i}v_{i}+x_{i}y_{i}$ with
$u_{i}v_{i}\in E(G_{i})$ and $x_{i}y_{i}\notin E(G_{i})$, where $G_{1}=G$.
Notice that some $\pm$-edge graphs of the $\pm e$-graph set $P_{\pm e}(G)$ are
not connected, or have cycles. Clearly, $S_{\pm e}(G)\subset P_{\pm e}(G)$,
and the subset $P_{\pm e}(G)\setminus S_{\pm e}(G)\neq\emptyset$, where the
$\pm e$-tree set $S_{\pm e}(G)$ is defined in Definition 24.

As $G^{c}$ is a colored $(q+1,q)$-tree admitting a vertex labeling
$\varphi:V(G^{c})\rightarrow[0,q]$ holding the vertex color set
$\varphi(V(G^{c}))=[0,q]$, we define the colored $\pm e$-graph set $P_{\pm
e}(G^{c})$ containing all colored $\pm$-edge graphs
$G^{c}_{i+1}=G^{c}_{i}-u_{i}v_{i}+x_{i}y_{i}$ with $u_{i}v_{i}\in
E(G^{c}_{i})$ and $x_{i}y_{i}\notin E(G^{c}_{i})$, where $G^{c}_{1}=G^{c}$.
Obviously, some $\pm$-edge graphs of the $\pm e$-graph set $P_{\pm e}(G^{c})$
are not trees, so we have $S_{\pm e}(G^{c})\subset P_{\pm e}(G^{c})$, and the
subset $P_{\pm e}(G^{c})\setminus S_{\pm e}(G^{c})\neq\emptyset$, where the
colored $\pm e$-tree set $S_{\pm e}(G^{c})$ is defined in Definition 25.
$\square$

######  Problem 11.

Let $N(t)$ be a connected network with $t\in[a,b]$, and let $P_{\pm e}(G)$ be
the $\pm e$-graph set defined in Definition 26. We define an $e$-total tree-
coloring $F$ for $N(t)$ as

$F:V(N(t))\rightarrow P_{\pm e}(G)\setminus S_{\pm e}(G),\quad
F:E(G)\rightarrow S_{\pm e}(G)$ (84)

with each edge $uv\in E(G)$ holding the $\pm e$-constraint

$\left\\{\begin{array}[]{cc}F(u)=F(uv)-a_{u}b_{u}+x_{u}y_{u},&a_{u}b_{u}\in
E(G),~{}x_{u}y_{u}\notin E(G)\\\
F(v)=F(uv)-a_{v}b_{v}+x_{v}y_{v},&a_{v}b_{v}\in E(G),~{}x_{v}y_{v}\notin
E(G)\end{array}\right.$ (85)

And we define a $v$-total tree-coloring $\theta$ for $N(t)$ as

$\theta:V(N(t))\rightarrow S_{\pm e}(G),\quad\theta:E(G)\rightarrow P_{\pm
e}(G)\setminus S_{\pm e}(G)$ (86)

with each edge $uv\in E(G)$ holding the $\pm e$-constraint

$\left\\{\begin{array}[]{cc}\theta(u)=\theta(uv)-s_{u}t_{u}+w_{u}z_{u},&s_{u}t_{u}\in
E(G),~{}w_{u}z_{u}\notin E(G)\\\
\theta(v)=\theta(uv)-s_{v}t_{v}+w_{v}z_{v},&s_{v}t_{v}\in
E(G),~{}w_{v}z_{v}\notin E(G)\end{array}\right.$ (87)

Describe the topological properties of the $\pm e$-graph set $P_{\pm e}(G)$ by
the connected network $N(t)$ admitting an $e$-total tree-coloring, or a
$v$-total tree-coloring, refer to [35].

######  Problem 12.

Let $N(t)$ be a connected network with $t\in[a,b]$, and let $P_{\pm e}(G^{c})$
be the colored $\pm e$-graph set defined in Definition 26. We define an
$e$-total colored-tree-coloring $f^{c}$ for $N(t)$ as

$f^{c}:V(N(t))\rightarrow P_{\pm e}(G^{c})\setminus S_{\pm e}(G^{c}),\quad
f^{c}:E(G^{c})\rightarrow S_{\pm e}(G^{c})$ (88)

with each edge $uv\in E(G^{c})$ holding the $\pm e$-constraint

$\left\\{\begin{array}[]{cc}f^{c}(u)=f^{c}(uv)-a_{u}b_{u}+x_{u}y_{u},&a_{u}b_{u}\in
E(G^{c}),~{}x_{u}y_{u}\notin E(G^{c})\\\
f^{c}(v)=f^{c}(uv)-a_{v}b_{v}+x_{v}y_{v},&a_{v}b_{v}\in
E(G^{c}),~{}x_{v}y_{v}\notin E(G^{c})\end{array}\right.$ (89)

And we define a $v$-total colored-tree-coloring $\varphi^{c}$ for $N(t)$ as

$\varphi^{c}:V(N(t))\rightarrow S_{\pm
e}(G^{c}),\quad\varphi^{c}:E(G^{c})\rightarrow P_{\pm e}(G^{c})\setminus
S_{\pm e}(G^{c})$ (90)

with each edge $uv\in E(G^{c})$ holding the $\pm e$-constraint

$\left\\{\begin{array}[]{cc}\varphi^{c}(u)=\varphi^{c}(uv)-s_{u}t_{u}+w_{u}z_{u},&s_{u}t_{u}\in
E(G^{c}),~{}w_{u}z_{u}\notin E(G^{c})\\\
\varphi^{c}(v)=\varphi^{c}(uv)-s_{v}t_{v}+w_{v}z_{v},&s_{v}t_{v}\in
E(G^{c}),~{}w_{v}z_{v}\notin E(G^{c})\end{array}\right.$ (91)

Characterize the topological structures of the $\pm e$-graph set $P_{\pm
e}(G^{c})$ by the connected network $N(t)$ admitting an $e$-total colored-
tree-coloring, or a $v$-total colored-tree-coloring, refer to [35]. It is
noticeable, each element $\varphi^{c}(w)=T^{c}_{i+1}$ of the Topcode-matrix
$T_{code}(N(t),\varphi^{c})$ corresponds to a Topcode-matrix
$T_{code}(T^{c}_{i+1},g_{T^{c}_{i+1}})$, like some thing in _Tensor_.

######  Problem 13.

We consider two $\pm e$-graph sets $P_{\pm e}(G)$ and $P_{\pm e}(G^{c})$ for a
uncolored connected $(p,q)$-graph $G$ and another colored connected
$(p,q)$-graph $G^{c}$ both having cycles, such that each graph of $P_{\pm
e}(G)$ is $G_{i+1}=G_{i}-u_{i}v_{i}+x_{i}y_{i}$ holding $u_{i}v_{i}\in
E(G_{i})$ and $x_{i}y_{i}\notin E(G_{i})$, where $G_{1}=G$, and each graph of
$P_{\pm e}(G^{c})$ is $G^{c}_{i+1}=G^{c}_{i}-u_{i}v_{i}+x_{i}y_{i}$ holding
$u_{i}v_{i}\in E(G^{c}_{i})$ and $x_{i}y_{i}\notin E(G^{c}_{i})$ with
$G^{c}_{1}=G^{c}$.

There is a $\pm e$-spanning-tree set $S_{\pm e}(T_{span})\subset P_{\pm e}(G)$
for each spanning tree $T_{span}$ of the connected $(p,q)$-graph $G$. Suppose
that the connected $(p,q)$-graph $G$ has $s(G)$ different spanning trees
$T^{1}_{span}$, $T^{2}_{span}$, $\dots$, $T^{s(G)}_{span}$, so we have the
$\pm e$-spanning-tree sets $S_{\pm e}(T^{i}_{span})\subset P_{\pm e}(G)$ with
$i\in[1,s(G)]$. Do we have $S_{\pm e}(T^{i}_{span})=S_{\pm e}(T^{j}_{span})$
for $1\leq i,j\leq s(G)$?

There is a colored $\pm e$-spanning-tree set $S_{\pm e}(T_{cspan})\subset
P_{\pm e}(G^{c})$ for each colored spanning tree $T_{cspan}$ of the colored
connected $(p,q)$-graph $G^{c}$. Suppose that the colored connected
$(p,q)$-graph $G^{c}$ has $s(G^{c})$ different colored spanning trees
$T^{1}_{cspan}$, $T^{2}_{cspan}$, $\dots$, $T^{s(G^{c})}_{cspan}$, so we have
the colored $\pm e$-spanning-tree sets $S_{\pm e}(T^{i}_{cspan})\subset P_{\pm
e}(G^{c})$ with $i\in[1,s(G^{c})]$. Is there $S_{\pm e}(T^{i}_{cspan})=S_{\pm
e}(T^{j}_{cspan})$ for $1\leq i,j\leq s(G^{c})$?

##  6 Topics related with parameterized colorings/labelings

###  6.1 Topcode-matrices having parameterized elements

Since a Topcode-matrix $T_{code}$ corresponds to a set $S_{gra}(T_{code})$ of
graphs, such that each graph $H\in S_{gra}(T_{code})$ has its own Topcode-
matrix $T_{code}(H)=T_{code}$, thereby, we will define $W$-constraint
$(k,d)$-total colorings by means of parameterized Topcode-matrices in the
following subsections. Number-based strings that accompanies parameterized
Topcode-matrices are useful in real application. We will introduce some
graphic lattices with the closure of some $W$-constraint $(k,d)$-total
colorings, and other coloring problems with parameters.

Part of operations and properties on Topcode-matrices, parameterized Topcode-
matrices and pan-Topcode-matrices (including set-type Topcode-matrices,
string-type Topcode-matrices, graph-type Topcode-matrices, real-valued
Topcode-matrices) have been introduced and investigated in [27], [33], [38],
and [40].

####  6.1.1 Algebraic operations on Topcode-matrices

In [38], the authors introduced the following operations on Topcode-matrices:

1\. Union-sum operation “$\uplus$”, union-sum Topcode-matrix and sub-Topcode-
matrix. Let $T^{i}_{code}=(X_{i},~{}E_{i}$, $Y_{i})^{T}_{3\times q_{i}}$ for
$i=1,2$ be Topcode-matrices, where

$X_{i}=(x^{i}_{1},x^{i}_{2},\dots,x^{i}_{q_{i}}),~{}E_{i}=(e^{i}_{1},e^{i}_{2},\dots,e^{i}_{q_{i}}),~{}Y_{i}=(y^{i}_{1},y^{i}_{2},\dots,y^{i}_{q_{i}}),~{}i=1,2$

The union operation “$\uplus$” of two Topcode-matrices $T^{1}_{code}$ and
$T^{2}_{code}$ is defined by

$T^{1}_{code}\uplus T^{2}_{code}=(X_{1}\uplus X_{2},~{}E_{1}\uplus
E_{2},~{}Y_{1}\uplus Y_{2})^{T}_{3\times(q_{1}+q_{2})}$ (92)

with

$X_{1}\uplus
X_{2}=(x^{1}_{1},x^{1}_{2},\dots,x^{1}_{q_{1}},x^{2}_{1},x^{2}_{2},\dots,x^{2}_{q_{2}})$,
$E_{1}\uplus
E_{2}=(e^{1}_{1},e^{1}_{2},\dots,e^{1}_{q_{1}},e^{2}_{1},e^{2}_{2},\dots,e^{2}_{q_{2}})$
and

$Y_{1}\uplus
Y_{2}=(y^{1}_{1},y^{1}_{2},\dots,y^{1}_{q_{1}},y^{2}_{1},y^{2}_{2},\dots,y^{2}_{q_{2}})$
and the process of obtaining the _union-sum Topcode-matrix_
$T^{1}_{code}\uplus T^{2}_{code}$ is called a _union operation_ , we call
$T^{i}_{code}$ to be a _sub-Topcode-matrix_ of the union-sum Topcode-matrix
$T^{1}_{code}\uplus T^{2}_{code}$, denoted as $T^{i}_{code}\subset
T^{1}_{code}\uplus T^{2}_{code}$ $i=1,2$.

Moreover, we can generalize Eq.(92) to the following form

$[\uplus]^{m}_{i=1}T^{i}_{code}=T^{1}_{code}\uplus
T^{2}_{code}\uplus\cdots\uplus
T^{m}_{code}=(X^{*},~{}E^{*},~{}Y^{*})^{T}_{3\times q^{*}}$ (93)

where

$X^{*}=X_{1}\uplus X_{2}\uplus\cdots\uplus X_{m}$, $E^{*}=E_{1}\uplus
E_{2}\uplus\cdots\uplus E_{m}$ and

$Y^{*}=Y_{1}\uplus Y_{2}\uplus\cdots\uplus Y_{m}$, as well as
$q^{*}=q_{1}+q_{2}+\cdots+q_{m}$ with $m\geq 2$.
And each $T^{i}_{code}$ is a _sub-Topcode-matrix_ of the union-sum Topcode-
matrix $[\uplus]^{m}_{i=1}T^{i}_{code}$.

2\. Subtraction operation “$\setminus$”. Let $T_{code}=T^{1}_{code}\uplus
T^{2}_{code}$ be the union-sum Topcode-matrix defined in Eq.(92). We remove
$T^{1}_{code}$ from $T_{code}$, and obtain

${\begin{split}T_{code}\setminus T^{1}_{code}&=(X_{1}\uplus X_{2}\setminus
X_{1},~{}E_{1}\uplus E_{2}\setminus E_{1},~{}Y_{1}\uplus Y_{2}\setminus
Y_{1})^{T}_{3\times(q_{1}+q_{2}-q_{1})}\\\
&=(X_{2},~{}E_{2},~{}Y_{2})^{T}_{3\times q_{2}}=T^{2}_{code}\end{split}}$ (94)

the process of obtaining $T^{2}_{code}$ from $T_{code}$ is called a
_subtraction operation_.

3\. Intersection operation “$\cap$”. Let
$T\,^{\prime}_{code}=T^{1}_{code}\uplus H_{code}$ and
$T\,^{\prime\prime}_{code}=T^{2}_{code}\uplus H_{code}$ be two Topcode-
matrices obtained by doing the union operation to Topcode-matrices
$T^{1}_{code},H_{code}$ and $T^{2}_{code}$. Since the Topcode-matrix
$H_{code}$ is a _common sub-Topcode-matrix_ of two Topcode-matrices
$T\,^{\prime}_{code}$ and $T\,^{\prime\prime}_{code}$, we take the largest
common sub-Topcode-matrix of $T\,^{\prime}_{code}$ and
$T\,^{\prime\prime}_{code}$ and denote it as $T\,^{\prime}_{code}\cap
T\,^{\prime\prime}_{code}$ called _intersected Topcode-matrix_ , this is so-
called the _intersection operation_ , so we have

$H_{code}\subseteq T\,^{\prime}_{code}\cap
T\,^{\prime\prime}_{code},~{}T\,^{\prime}_{code}\cap
T\,^{\prime\prime}_{code}\subseteq
T\,^{\prime}_{code},~{}T\,^{\prime}_{code}\cap
T\,^{\prime\prime}_{code}\subseteq T\,^{\prime\prime}_{code}$ (95)

4\. Union operation “$\cup$”. For the intersected Topcode-matrix
$T\,^{\prime}_{code}\cap T\,^{\prime\prime}_{code}$ based on two Topcode-
matrices $T\,^{\prime}_{code}$ and $T\,^{\prime\prime}_{code}$, we define the
union operation “$\cup$” on $T\,^{\prime}_{code}$ and
$T\,^{\prime\prime}_{code}$ as

$T\,^{\prime}_{code}\cup
T\,^{\prime\prime}_{code}=T\,^{\prime}_{code}\uplus[T\,^{\prime\prime}_{code}\setminus(T\,^{\prime}_{code}\cap
T\,^{\prime\prime}_{code})]=[T\,^{\prime}_{code}\setminus(T\,^{\prime}_{code}\cap
T\,^{\prime\prime}_{code})]\uplus T\,^{\prime\prime}_{code}$ (96)

5\. Coinciding operation “$\odot$”. Let
$T\,^{\prime}_{code}=T^{1}_{code}\uplus H_{code}$ and
$T\,^{\prime\prime}_{code}=T^{2}_{code}\uplus H_{code}$ be two Topcode-
matrices obtained by doing the union operation to Topcode-matrices
$T^{1}_{code},H_{code}$ and $T^{2}_{code}$. So $H_{code}\subset
T\,^{\prime}_{code}$ and $H_{code}\subset T\,^{\prime\prime}_{code}$. We
define the _coinciding operation_ “$\odot$” on two Topcode-matrices
$T\,^{\prime}_{code}$ and $T\,^{\prime\prime}_{code}$ as follows

$[\odot H_{code}]\langle
T\,^{\prime}_{code},T\,^{\prime\prime}_{code}\rangle=T^{1}_{code}\uplus
H_{code}\uplus T^{2}_{code}=[T\,^{\prime}_{code}\setminus H_{code}]\uplus
T\,^{\prime\prime}_{code}=[T\,^{\prime\prime}_{code}\setminus H_{code}]\uplus
T\,^{\prime}_{code}$ (97)

6\. Splitting operation “$\wedge$”. We do the splitting operation to the sub-
Topcode-matrix $H_{code}$ of a coincided Topcode-matrix $T^{1}_{code}\uplus
H_{code}\uplus T^{2}_{code}$, such that the resultant Topcode-matrix consisted
of two disjoint Topcode-matrices $T^{1}_{code}\uplus H_{code}$ and
$T^{2}_{code}\uplus H_{code}$, denoted as

$\\{[\odot H_{code}]\langle
T\,^{\prime}_{code},T\,^{\prime\prime}_{code}\rangle\\}\wedge
H_{code}=[T^{1}_{code}\uplus H_{code}\uplus T^{2}_{code}]\wedge H_{code}$ (98)

######  Example 6.

For the following three Topcode-matrices

$\centering{\begin{split}T_{code}=\left(\begin{array}[]{ccccc}x_{1}&x_{2}&\cdots&x_{n}\\\
e_{1}&e_{2}&\cdots&e_{n}\\\
y_{1}&y_{2}&\cdots&y_{n}\end{array}\right),~{}T^{i}_{code}=\left(\begin{array}[]{ccccc}x^{i}_{1}&x^{i}_{2}&\cdots&x^{i}_{m}\\\
e^{i}_{1}&e^{i}_{2}&\cdots&e^{i}_{m}\\\
y^{i}_{1}&y^{i}_{2}&\cdots&y^{i}_{m}\end{array}\right),~{}i=1,2\end{split}}\@add@centering$
(99)

we have two union-sum Topcode-matrices

$\centering{\begin{split}T_{code}\uplus
T^{i}_{code}=\left(\begin{array}[]{cccccccc}x_{1}&x_{2}&\cdots&x_{n}&x^{i}_{1}&x^{i}_{2}&\cdots&x^{i}_{m}\\\
e_{1}&e_{2}&\cdots&e_{n}&e^{i}_{1}&e^{i}_{2}&\cdots&e^{i}_{m}\\\
y_{1}&y_{2}&\cdots&y_{n}&y^{i}_{1}&y^{i}_{2}&\cdots&y^{i}_{m}\end{array}\right),~{}i=1,2\end{split}}\@add@centering$
(100)

A coincided Topcode-matrix is

$J_{code}=[\odot T^{2}_{code}]\langle T_{code}\uplus
T^{2}_{code},T^{1}_{code}\uplus T^{2}_{code}\rangle$ (101)

and

$\centering{\begin{split}J_{code}=\left(\begin{array}[]{cccccccccccc}x_{1}&x_{2}&\cdots&x_{n}&x^{2}_{1}&x^{2}_{2}&\cdots&x^{2}_{s}&x^{1}_{1}&x^{1}_{2}&\cdots&x^{1}_{m}\\\
e_{1}&e_{2}&\cdots&e_{n}&e^{2}_{1}&e^{2}_{2}&\cdots&e^{2}_{s}&e^{1}_{1}&e^{1}_{2}&\cdots&e^{1}_{m}\\\
y_{1}&y_{2}&\cdots&y_{n}&y^{2}_{1}&y^{2}_{2}&\cdots&y^{2}_{s}&y^{1}_{1}&y^{1}_{2}&\cdots&y^{1}_{m}\end{array}\right)=T_{code}\uplus
T^{2}_{code}\uplus T^{1}_{code}\end{split}}\@add@centering$ (102)

######  Example 7.

Given two Topcode-matrices

$\centering
A=\left(\begin{array}[]{ccccccccc}\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&5&7&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&3&5&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&18&14&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}\end{array}\right),\quad
B=\left(\begin{array}[]{ccccccccc}\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&5\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&9\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&12\end{array}\right)\@add@centering$
(103)

we have

$\centering A\uplus
B=\left(\begin{array}[]{ccccccccc}\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&5&7&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&5\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&3&5&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&9\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&18&14&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&12\end{array}\right),~{}A\cup
B=\left(\begin{array}[]{ccccccccc}\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&5&7&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&5\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&3&5&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&9\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&18&14&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&12\end{array}\right)\@add@centering$
(104)

$\centering~{}B\setminus A=\left(\begin{array}[]{ccccccccc}5\\\ 9\\\
12\end{array}\right),~{}A\cap
B=\left(\begin{array}[]{ccccccccc}\textbf{{\color[rgb]{0.00,0.00,1.00}7}}&\textbf{{\color[rgb]{0.00,0.00,1.00}1}}\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}1}}&\textbf{{\color[rgb]{0.00,0.00,1.00}7}}\\\
\textbf{{\color[rgb]{0.00,0.00,1.00}18}}&\textbf{{\color[rgb]{0.00,0.00,1.00}18}}\end{array}\right),~{}A\setminus
B=\left(\begin{array}[]{ccccccccc}5&7\\\ 3&5\\\
18&14\end{array}\right)\@add@centering$ (105)

The split Topcode-matrix $(A\cup B)\wedge(A\cap B)$ consists of $A$ and $B$,
and there are the following Topcode-matrix algebraic relationships:

${\begin{split}&A=(A\uplus B)\setminus B,~{}B=(A\uplus B)\setminus
A,~{}(A\setminus B)\uplus(B\setminus A)=(A\cup B)\setminus(A\cap B)\\\ &A\cup
B=(A\setminus B)\uplus(B\setminus A)\uplus(A\cap B),~{}A\cup B=[\odot(A\cap
B)]\langle A,B\rangle\\\ &[\odot(A\cap B)]\langle A,B\rangle=(A\setminus
B)\uplus B=A\uplus(B\setminus A)\end{split}}$ (106)

About the colored Topcode-matrix defined in Definition 1 and the colored
Topcode-matrix defined in Definition 2, we have some operations of Topcode-
matrix algebra as follows:

  1. ATcode-1.

The values of $x_{i}$, $e_{i}$ and $y_{i}$ with $i\in[1,q]$ can be any things
in the world. We say $e_{i}$ to be _valued_ by $x_{i}$ and $y_{i}$ if there
exists a _valued function_ $f$ defined as $e_{i}=f(x_{i},y_{i})$ for
$i\in[1,q]$, and furthermore $E=f(X,Y)$.

  2. ATcode-2.

The graph $G$ has another _Topcode-matrix_ $T\,^{XY}(G)=(X,Y)^{T}_{2\times
q}$.

  3. ATcode-3.

In the view of graph colorings, we regard that $T\,^{XY}(G)$ corresponds to a
_vertex coloring_ of $G$, $E=(e_{1},e_{2},\dots,e_{q})$ corresponds to an
_edge coloring_ of $G$, and $T_{code}(G)$ corresponds to a _total coloring_ of
$G$.

  4. ATcode-4.

A Topcode-matrix $T_{code}(G)$ may be the common Topcode-matrix of many graphs
$G_{1},G_{2},\dots,G_{n}$, that is $T_{code}(G)=T_{code}(G_{i})$ with
$i\in[1,n]$.

  5. ATcode-5.

A _column exchange_ $C_{(i,j)}$ of vertex-vectors and edge-vector of a
Topcode-matrix $T_{code}$ is defined as:

${\begin{split}C_{(i,j)}(X)&=C_{(i,j)}(x_{1},x_{2},\dots,x_{i-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{x_{i}}},x_{i+1},\dots,x_{j-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{x_{j}}},x_{j+1},\dots,x_{q})\\\
&=(x_{1},x_{2},\dots,x_{i-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{x_{j}}},x_{i+1},\dots,x_{j-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{x_{i}}},x_{j+1},\dots,x_{q})\\\
C_{(i,j)}(E)&=C_{(i,j)}(e_{1},e_{2},\dots,e_{i-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{e_{i}}},e_{i+1},\dots,e_{j-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{e_{j}}},e_{j+1},\dots,e_{q})\\\
&=(e_{1},e_{2},\dots,e_{i-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{e_{j}}},e_{i+1},\dots,e_{j-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{e_{i}}},e_{j+1},\dots,e_{q})\\\
C_{(i,j)}(X)&=C_{(i,j)}(y_{1},y_{2},\dots,y_{i-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{y_{i}}},y_{i+1},\dots,y_{j-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{y_{j}}},y_{j+1},\dots,y_{q})\\\
&=(y_{1},y_{2},\dots,y_{i-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{y_{j}}},y_{i+1},\dots,y_{j-1},{\color[rgb]{0.98,0.00,0.00}\mathbf{y_{i}}},y_{j+1},\dots,y_{q}),\end{split}}$
(107)

so we get a column exchange of a Topcode-matrix $T_{code}$ as:

$C_{(i,j)}(T_{code})=C_{(i,j)}(X,E,Y)^{T}=(C_{(i,j)}(X),C_{(i,j)}(E),C_{(i,j)}(Y))^{T}$
(108)

For two Topcode-matrices $T\,^{r}_{code}$ and $T\,^{s}_{code}$, if there are
column exchanges $C_{(i_{k},j_{k})}$ with $k\in[1,m]$, such that

$C_{(i_{m},j_{m})}C_{(i_{m-1},j_{m-1})}\cdots
C_{(i_{1},j_{1})}(T\,^{r}_{code})=T\,^{s}_{code}$ (109)

we say that both Topcode-matrices $T\,^{r}_{code}$ and $T\,^{s}_{code}$ are
similar from each other, write this fact as

$T\,^{r}_{code}(\sim)T\,^{s}_{code}$ (110)

  6. ATcode-6.

A _line exchange_ $L_{(i)}(T_{code})$ on the vertex-vectors of a Topcode-
matrix $T_{code}$ is defined as:

$\left(\begin{array}[]{ccccc}x_{1}\\\ e_{1}\\\
y_{1}\end{array}\right)\bigcup\cdots\bigcup\left(\begin{array}[]{ccccc}{\color[rgb]{0.98,0.00,0.00}\mathbf{x_{i}}}\\\
e_{i}\\\
{\color[rgb]{0.98,0.00,0.00}\mathbf{y_{i}}}\end{array}\right)\bigcup\cdots\bigcup\left(\begin{array}[]{ccccc}x_{q}\\\
e_{q}\\\
y_{q}\end{array}\right)\rightarrow\left(\begin{array}[]{ccccc}x_{1}\\\
e_{1}\\\
y_{1}\end{array}\right)\bigcup\cdots\bigcup\left(\begin{array}[]{ccccc}{\color[rgb]{0.98,0.00,0.00}\mathbf{y_{i}}}\\\
e_{i}\\\
{\color[rgb]{0.98,0.00,0.00}\mathbf{x_{i}}}\end{array}\right)\bigcup\cdots\bigcup\left(\begin{array}[]{ccccc}x_{q}\\\
e_{q}\\\ y_{q}\end{array}\right)$ (111)

denoted as $L_{(i)}(T_{code})=(L_{(i)}(X),E,L_{(i)}(Y))^{T}$. For two Topcode-
matrices $T\,^{r}_{code}$ and $T\,^{s}_{code}$, if there are line exchanges
$L_{(i_{k})}$ with $k\in[1,m]$, such that

$L_{(i_{m})}L_{(i_{m-1})}\cdots L_{(i_{1})}(T\,^{r}_{code})=T\,^{s}_{code}$
(112)

then both Topcode-matrices $T\,^{r}_{code}$ and $T\,^{s}_{code}$ are similar
from each other, that is $T\,^{r}_{code}(\sim)T\,^{s}_{code}$.

######  Example 8.

There two Topcode-matrices as follows:

$\centering{\begin{split}T\,^{a}_{code}=\left(\begin{array}[]{ccccccc}10&7&0&0&2&2&0\\\
1&3&5&7&9&11&13\\\
11&10&5&7&11&13&13\end{array}\right),~{}T\,^{b}_{code}=\left(\begin{array}[]{ccccccc}0&10&10&0&7&2&2\\\
13&1&3&5&7&9&11\\\
13&11&7&5&0&11&13\end{array}\right)\end{split}}\@add@centering$ (113)

We do two line exchanges $L_{(4)}$ and $L_{(2)}$ to $T\,^{a}_{code}$, and do a
column exchange $C_{(i,j)}$ to $T\,^{a}_{code}$, so we get a _mixed operation_
$C_{(i,j)}L_{(4)}L_{(2)}(T\,^{a}_{code})=T\,^{b}_{code}$, that is
$T\,^{a}_{code}(\sim)T\,^{b}_{code}$. $\square$

  7. ATcode-7.

The coinciding-operation and the joining-operation on Topcode-matrices. Let
$T_{code}(G)=(X,E,Y)^{T}$ be the Topcode-matrix of a $(p,q)$-graph $G$, and
let $T_{code}(H)=(X\,^{\prime}$, $E\,^{\prime}$, $Y\,^{\prime})^{T}$ be the
Topcode-matrix of a $(p,q)$-graph $H$, and let $G$ and $H$ be disjoint from
each other, that is $X\cap X\,^{\prime}=\emptyset$, $E\cap
E\,^{\prime}=\emptyset$ and $Y\cap Y\,^{\prime}=\emptyset$. If we vertex-
coincide the vertex $x_{i}$ of $G$ with the vertex $x\,^{\prime}_{j}$ of $H$
into one vertex $w=x_{i}\odot x\,^{\prime}_{j}$, then the vertex-coincided
graph $G\odot H$ has its own Topcode-matrix as follows

$T_{code}(G\odot_{w}H)=T^{*}_{code}(G)\bigcup T^{*}_{code}(H)$ (114)

called _Topcode-matrix coinciding-operation_ , where the Topcode-matrix
$T^{*}_{code}(G)$ is the result of replacing each $x_{i}$ by $w$ in
$T_{code}(G)$, and the Topcode-matrix $T^{*}_{code}(H)$ is obtain by replacing
each $x\,^{\prime}_{j}$ by $w$ in $T_{code}(H)$. Next, we add a new edge
$e=x_{i}x\,^{\prime}_{j}$ to join the vertex $x_{i}$ of $G$ with the vertex
$x\,^{\prime}_{j}$ of $H$ together, such that the edge-joined graph $G\ominus
H$ has the Topcode-matrix defined as

$T_{code}(G\ominus
H)=T_{code}(G)\bigcup(x_{i},~{}e,~{}x\,^{\prime}_{j})^{T}\bigcup T_{code}(H)$
(115)

called _Topcode-matrix joining-operation_.

  8. ATcode-8.

Operations on meta-matrices.

(A) Let $T\,^{i}_{3\times 1}=(x_{i},e_{i},y_{i})^{T}$ be a _meta-matrix_ of a
Topcode-matrix $T_{code}$ defined in Definition 1. We define the _union
operation_ of meta-matrices as

$T_{code}=\bigcup^{q}_{i=1}T\,^{i}_{3\times
1}=\left(\begin{array}[]{ccccc}x_{1}\\\ e_{1}\\\
y_{1}\end{array}\right)\bigcup\left(\begin{array}[]{ccccc}x_{2}\\\ e_{2}\\\
y_{2}\end{array}\right)\bigcup\cdots\bigcup\left(\begin{array}[]{ccccc}x_{q}\\\
e_{q}\\\ y_{q}\end{array}\right)=\bigcup^{q}_{i=1}(x_{i},e_{i},y_{i})^{T}$
(116)

(B) Suppose that $x_{i},e_{i}$ and $y_{i}$ of each meta-matrix
$T\,^{i}_{3\times 1}$ with $i\in[1,q]$ are integers. The _meta-matrix number
multiplication_ is defined as

$\beta\cdot T\,^{i}_{3\times 1}=(\beta\cdot x_{i},\beta\cdot e_{i},\beta\cdot
y_{i})^{T}$ (117)

with an integer $\beta\geq 1$, and the _meta-matrix add-operation_ is defined
as

$\alpha\cdot T\,^{i}_{3\times 1}+\beta\cdot T\,^{j}_{3\times 1}=(\alpha\cdot
x_{i}+\beta\cdot x_{j},\alpha\cdot e_{i}+\beta\cdot e_{j},\alpha\cdot
y_{i}+\beta\cdot y_{j})^{T}$ (118)

with integer $\alpha,\beta\geq 1$.

(C) Meta-matrix lattices. Let $\textbf{{T}}=(T\,^{1}_{3\times
1},T\,^{2}_{3\times 1},\dots,T\,^{q}_{3\times 1})$ be a _meta-matrix base_
consisted of meta-matrices $T\,^{i}_{3\times 1}$ for $i\in[1,q]$. Notice that
$T\,^{i}_{3\times 1}\cap T\,^{j}_{3\times 1}=\emptyset$ for $i\neq j$, so
$T\,^{i}_{3\times 1}\cup T\,^{j}_{3\times 1}=T\,^{i}_{3\times 1}\uplus
T\,^{j}_{3\times 1}$. We call the following set

$\textrm{{L}}(\cup\textbf{{T}})=\left\\{\bigcup^{q}_{i=1}\beta_{k}\cdot
T\,^{k}_{3\times 1}:\beta_{k}\in Z^{0},T\,^{k}_{3\times
1}\in\textbf{{T}}\right\\}=\left\\{[\uplus]^{q}_{i=1}\beta_{k}\cdot
T\,^{k}_{3\times 1}:\beta_{k}\in Z^{0},T\,^{k}_{3\times
1}\in\textbf{{T}}\right\\}$ (119)

a _meta-matrix lattice_ based on the meta-matrix base
$\textbf{{T}}=(T\,^{1}_{3\times 1},T\,^{2}_{3\times 1},\dots,T\,^{q}_{3\times
1})$. In general, some matrix $\bigcup^{q}_{i=1}\beta_{k}T\,^{k}_{3\times
1}\in\textrm{{L}}(\cup\textbf{{T}})$ is not a Topcode-matrix of any graph.

Since there are $q!$ permutations for each one of $x_{1}x_{2}\cdots x_{q}$,
$e_{1}e_{2}\cdots e_{q}$ and $y_{1}y_{2}\cdots y_{q}$, by the column exchange
operation on Topcode-matrices, we have

######  Theorem 37.

Any Topcode-matrix $T_{code}$ of order $3\times q$ is similar with other
$(q!-1)$ Topcode-matrices of order $3\times q$.

######  Theorem 38.

If there are different graphs $G,G_{1},G_{2},\dots,G_{n}$ holding
$T_{code}(G)=T_{code}(G_{i})$ and $|V(G)|\leq|V(G_{i})|$ for $i\in[1,n]$, then
each graph $G_{i}$ admits a graph homomorphism to $G$, i.e. $G_{i}\rightarrow
G$ for $i\in[1,n]$.

####  6.1.2 Parameterized Topcode-matrices

Matrices having parameterized elements are called $(k,d)$-type matrices, for
example, $(k,d)$-graceful matrices, $(k,d)$-edge-magic matrices,
$(k,d)$-harmonious matrices, in the following discussion.

######  Definition 27.

∗ A _parameterized Topcode-matrix_ $P_{ara}=(X_{k,d},E_{k,d},Y_{k,d})^{T}$
shown in Eq.(120) is defined as

$X_{k,d}=(\alpha_{1}k+a_{1}d,\alpha_{2}k+a_{2}d,\dots,\alpha_{q}k+a_{q}d)$,

$Y_{k,d}=(\beta_{1}k+b_{1}d,\beta_{2}k+b_{2}d,\dots,\beta_{q}k+b_{q}d)$, and

$E_{k,d}=(\gamma_{1}k+c_{1}d,\gamma_{2}k+c_{2}d,\dots,\gamma_{q}k+c_{q}d)$
for two integers $k,d\geq 1$, $\square$

Moreover we can partition a parameterized Topcode-matrix into the linear
combination of Topcode-matrices as follows:

${\begin{split}P_{ara}&=\left(\begin{array}[]{ccccc}\alpha_{1}k+a_{1}d&\alpha_{2}k+a_{2}d&\cdots&\alpha_{q}k+a_{q}d\\\
\gamma_{1}k+c_{1}d&\gamma_{2}k+c_{2}d&\cdots&\gamma_{q}k+c_{q}d\\\
\beta_{1}k+b_{1}d&\beta_{2}k+b_{2}d&\cdots&\beta_{q}k+b_{q}d\end{array}\right)_{3\times
q}\\\
&=\left(\begin{array}[]{ccccc}\alpha_{1}k&\alpha_{2}k&\cdots&\alpha_{q}k\\\
\gamma_{1}k&\gamma_{2}k&\cdots&\gamma_{q}k\\\
\beta_{1}k&\beta_{2}k&\cdots&\beta_{q}k\end{array}\right)_{3\times
q}+\left(\begin{array}[]{ccccc}a_{1}d&a_{2}d&\cdots&a_{q}d\\\
c_{1}d&c_{2}d&\cdots&c_{q}d\\\
b_{1}d&b_{2}d&\cdots&b_{q}d\end{array}\right)_{3\times q}\\\ &=k\cdot
I_{3\times q}+d\cdot C_{3\times q}\end{split}}$ (120)

where $I_{3\times q}$ is called _constant Topcode-matrix_ defined as

${\begin{split}I_{3\times
q}=\left(\begin{array}[]{ccccc}\alpha_{1}&\alpha_{2}&\cdots&\alpha_{q}\\\
\gamma_{1}&\gamma_{2}&\cdots&\gamma_{q}\\\
\beta_{1}&\beta_{2}&\cdots&\beta_{q}\end{array}\right)_{3\times
q}=(I_{X},I_{E},I_{Y})^{T}\end{split}}$ (121)

with three vectors $I_{X}=(\alpha_{1},\alpha_{2},\dots,\alpha_{q})$,
$I_{E}=(\gamma_{1},\gamma_{2},\dots,\gamma_{q})$ and
$I_{Y}=(\beta_{1},\beta_{2},\dots,\beta_{q})$; and $C_{3\times q}$ is, called
_main Topcode-matrix_ defined as

${\begin{split}C_{3\times
q}=\left(\begin{array}[]{ccccc}a_{1}&a_{2}&\cdots&a_{q}\\\
c_{1}&c_{2}&\cdots&c_{q}\\\
b_{1}&b_{2}&\cdots&b_{q}\end{array}\right)_{3\times
q}=(C_{X},C_{E},C_{Y})^{T}\end{split}}$ (122)

for three vectors $C_{X}=(a_{1},a_{2},\dots,a_{q})$,
$C_{E}=(c_{1},c_{2},\dots,c_{q})$, and $C_{Y}=(b_{1},b_{2},\dots,b_{q})$.
Also, the parameterized Topcode-matrix can be written as

$P_{ara}=k\cdot(I_{X},I_{E},I_{Y})^{T}+d\cdot(C_{X},C_{E},C_{Y})^{T}=k\cdot
I_{3\times q}+d\cdot C_{3\times q}.$ (123)

In the valued case, there is a _valued function_ $\theta$ to combine
$C_{E},C_{X}$ and $C_{Y}$ together by $C_{E}=\theta(C_{X},C_{Y})$ with
$c_{i}=W\langle a_{i},b_{i}\rangle$ for $i\in[1,q]$ under a $W$-constraint.

For _bipartite graphs_ , especially, we define the _unite Topcode-matrix_ as
follows

${\begin{split}I\,^{0}=\left(\begin{array}[]{ccccc}0&0&\cdots&0\\\
1&1&\cdots&1\\\ 1&1&\cdots&1\end{array}\right)_{3\times
q}=(X\,^{0},~{}E\,^{0},~{}Y\,^{0})^{T}\end{split}}$ (124)

with two vertex-vectors $X\,^{0}=(0,0,\dots,0)_{1\times q}$ and
$Y\,^{0}=(1,1,\dots,1)_{1\times q}$, and the edge-vector
$E\,^{0}=(1,1,\dots,1)_{1\times q}$.

######  Example 9.

In Eq.(120), if we have an edge-difference constraint

$(\gamma_{i}k+c_{i}d)+|(\beta_{i}k+b_{i}d)-(\alpha_{i}k+a_{i}d)|=(\gamma_{i}+\beta_{i}-\alpha_{i})k+(c_{i}+b_{i}-a_{i})d=k\cdot
A_{ed}+d\cdot B_{ed}$

for $i\in[1,q]$, we write the above fact as $I_{E}+|I_{Y}-I_{X}|:=A_{ed}$ and
$C_{E}+|C_{Y}-C_{X}|:=B_{ed}$, where $A_{ed}$ and $B_{ed}$ are constants.

If there is an edge-magic constraint

$(\gamma_{i}k+c_{i}d)+(\beta_{i}k+b_{i}d)+(\alpha_{i}k+a_{i}d=(\gamma_{i}+\beta_{i}+\alpha_{i})k+(c_{i}+b_{i}+a_{i})d)=k\cdot
A_{em}+d\cdot B_{em}$

for $i\in[1,q]$, we write the above fact as $I_{X}+I_{E}+I_{Y}:=A_{em}$ and
$C_{X}+C_{E}+C_{Y}:=B_{em}$ for constants $A_{em}$ and $B_{em}$.

If we have a graceful-difference constraint

$\big{|}|(\beta_{i}k+b_{i}d)-(\alpha_{i}k+a_{i}d)|-(\gamma_{i}k+c_{i}d)\big{|}=(\beta_{i}-\alpha_{i}-\gamma_{i})k+(b_{i}-a_{i}-c_{i})d=k\cdot
A_{gd}+d\cdot B_{gd}$

for $i\in[1,q]$, we write the above fact as
$\big{|}|I_{Y}-I_{X}|-I_{E}\big{|}:=A_{gd}$ and
$\big{|}|C_{Y}-C_{X}|-C_{E}\big{|}:=B_{gd}$ for constants $A_{gd}$ and
$B_{gd}$.

If we have felicitous-difference constraint

$|(\beta_{i}k+b_{i}d)+(\alpha_{i}k+a_{i}d)-(\gamma_{i}k+c_{i}d)|=(\beta_{i}+\alpha_{i}-\gamma_{i})k+(b_{i}+a_{i}-c_{i})d=k\cdot
A_{fd}+d\cdot B_{fd}$

for $i\in[1,q]$, we write the above fact as $|I_{Y}+I_{X}-I_{E}|:=A_{fd}$ and
$|C_{Y}+C_{X}-C_{E}|:=B_{fd}$ for constants $A_{fd}$ and $B_{fd}$. $\square$

If there is no confusion, we omit “’order $3\times q$’ in the following
discussion, or add a sentence “ the Topcode-matrices $I$, $T_{code}(G)$ and
$P_{ara}(G)$ have the same order”.

######  Definition 28.

∗ Since a Topcode-matrix $T_{code}(G)$ defined in Eq.(17) for a $(p,q)$-graph
$G$ can be as a main Topcode-matrix in Eq.(122), so we get two parameterized
Topcode-matrices

$P_{ara}(G)=k\cdot I+d\cdot T_{code}(G),\quad P_{ara}(G,F)=k\cdot
I\,^{0}+d\cdot T_{code}(G,f)$ (125)

where the Topcode-matrices $I$, $T_{code}(G)$ and $P_{ara}(G)$, $I\,^{0}$,
$T_{code}(G,f)$ and $P_{ara}(G,F)$ have the same order, and $f$ is a
$W$-constraint coloring admitted by $G$, as well as $F$ is a $W$-constraint
parameterized coloring of $G$. $\square$

######  Example 10.

In Fig.41, there are four colored bipartite graphs $G_{1},G_{2},G_{3},G_{4}$,
their $(k,d)$-type Topcode-matrices $P_{ara}(G_{i},F_{i})$ with $i\in[1,4]$
are shown in Eq.(127) and Eq.(128) respectively. By the unite Topcode-matrix
$I\,^{0}$ defined in Eq.(124) and Definition 28, we rewrite these $(k,d)$-type
Topcode-matrices by

$P_{ara}(G_{j},F_{j})=k\cdot I\,^{0}+d\cdot
T_{code}(G\,^{\prime}_{j},g_{j}),~{}j\in[1,4]$ (126)

for integers $k\geq 0$ and $d\geq 1$, where the Topcode-matrices $I\,^{0}$,
$T_{code}(G\,^{\prime}_{j},g_{j})$ and $P_{ara}(G_{j},F_{j})$ have the same
order $3\times 9$, four Topcode-matrices $T_{code}(G\,^{\prime}_{j},g_{j})$
for $j\in[1,4]$ are shown in Eq.(129) and Eq.(130), respectively, and moreover

(1) $P_{ara}(G_{1},F_{1})$ is a _$(k,d)$ -graceful Topcode-matrix_, where
$G_{1}$ admits a $(k,d)$-graceful labeling $F_{1}$ and $G\,^{\prime}_{1}$
admits a graceful labeling $g_{1}$ corresponding to a Topcode-matrix
$T_{code}(G\,^{\prime}_{1},g_{1})$;

(2) $P_{ara}(G_{2},F_{2})$ is a _$(k,d)$ -felicitous Topcode-matrix_, where
$G_{2}$ admits a $(k,d)$-felicitous labeling $F_{2}$ and $G\,^{\prime}_{2}$
admits a felicitous labeling $g_{2}$ corresponding to a Topcode-matrix
$T_{code}(G\,^{\prime}_{2},g_{2})$;

(3) $P_{ara}(G_{3},F_{3})$ is a _$(k,d)$ -edge-magic Topcode-matrix_, where
$G_{3}$ admits a $(k,d)$-edge-magic labeling $F_{3}$ and $G\,^{\prime}_{3}$
admits an edge-magic labeling $g_{3}$ corresponding to a Topcode-matrix
$T_{code}(G\,^{\prime}_{3},g_{3})$;

(4) $P_{ara}(G_{4},F_{4})$ is a _$(k,d)$ -edge-antimagic Topcode-matrix_,
where $G_{4}$ admits a $(k,d)$-edge-antimagic labeling $F_{4}$ and
$G\,^{\prime}_{4}$ admits an edge antimagic labeling $g_{4}$ corresponding to
a Topcode-matrix $T_{code}(G\,^{\prime}_{4}$, $g_{4})$. $\square$

$\centering{\begin{split}P_{ara}(G_{1},F_{1})&=\left(\begin{array}[]{ccccccccccccccc}2d&2d&d&2d&d&0&d&0&0\\\
k&k+d&k+2d&k+3d&k+4d&k+5d&k+6d&k+7d&k+8d\\\
k+2d&k+3d&k+3d&k+5d&k+5d&k+5d&k+7d&k+7d&k+8d\end{array}\right)\\\
P_{ara}(G_{2},F_{2})&=\left(\begin{array}[]{ccccccccccccccc}d&2d&2d&0&0&d&0&d&2d\\\
k&k+d&k+2d&k+3d&k+4d&k+5d&k+6d&k+7d&k+8d\\\
k+8d&k+8d&k+9d&k+3d&k+4d&k+4d&k+6d&k+6d&k+6d\end{array}\right)\end{split}}\@add@centering$
(127)

$\centering{\begin{split}P_{ara}(G_{3},F_{3})&=\left(\begin{array}[]{ccccccccccccccc}3d&3d&2d&3d&2d&2d&2d&d&0\\\
k&k+d&k+2d&k+3d&k+4d&k+5d&k+6d&k+7d&k+8d\\\
k+8d&k+7d&k+7d&k+5d&k+5d&k+4d&k+3d&k+3d&k+3d\end{array}\right)\\\
P_{ara}(G_{4},F_{4})&=\left(\begin{array}[]{ccccccccccccccc}0&0&d&0&d&2d&d&2d&2d\\\
k&k+d&k+2d&k+3d&k+4d&k+5d&k+6d&k+7d&k+8d\\\
k+2d&k+3d&k+3d&k+5d&k+5d&k+5d&k+7d&k+7d&k+8d\end{array}\right)\end{split}}\@add@centering$
(128)

Figure 41: Four Topsnut-gpws corresponding to four $(k,d)$-type Topcode-
matrices shown in Eq.(127).

$T_{code}(G\,^{\prime}_{1},g_{1})=\left(\begin{array}[]{ccccccccccccccc}2&2&1&2&1&0&1&0&0\\\
0&1&2&3&4&5&6&7&8\\\
2&3&3&5&5&5&7&7&8\end{array}\right)~{}T_{code}(G\,^{\prime}_{2},g_{2})=\left(\begin{array}[]{ccccccccccccccc}1&2&2&0&0&1&0&1&2\\\
0&1&2&3&4&5&6&7&8\\\ 8&8&9&3&4&4&6&6&6\end{array}\right)$ (129)
$T_{code}(G\,^{\prime}_{3},g_{3})=\left(\begin{array}[]{ccccccccccccccc}3&3&2&3&2&2&2&1&0\\\
0&1&2&3&4&5&6&7&8\\\
8&7&7&5&5&4&3&3&3\end{array}\right)~{}T_{code}(G\,^{\prime}_{4},g_{4})=\left(\begin{array}[]{ccccccccccccccc}0&0&1&0&1&2&1&2&2\\\
0&1&2&3&4&5&6&7&8\\\ 2&3&3&5&5&5&7&7&8\end{array}\right)$ (130)

######  Definition 29.

[42] A total labeling $f:V(G)\cup E(G)\rightarrow[1,p+q]$ for a bipartite
$(p,q)$-graph $G$ is a bijection and holds:

(i) (e-magic) $f(uv)+|f(u)-f(v)|=k$;

(ii) (ee-difference) each edge $uv$ matches with another edge $xy$ holding one
of $f(uv)=|f(x)-f(y)|$ and $f(uv)=2(p+q)-|f(x)-f(y)|$ true;

(iii) (ee-balanced) let $s(uv)=|f(u)-f(v)|-f(uv)$ for each edge $uv\in E(G)$,
then there exists a constant $k\,^{\prime}$ such that each edge $uv$ matches
with another edge $u\,^{\prime}v\,^{\prime}$ holding one of
$s(uv)+s(u\,^{\prime}v\,^{\prime})=k\,^{\prime}$ and
$2(p+q)+s(uv)+s(u\,^{\prime}v\,^{\prime})=k\,^{\prime}$ true;

(iv) (EV-ordered) $\min f(V(G))>\max f(E(G))$, or $\max f(V(G))<\min f(E(G))$,
or $f(V(G))\subseteq f(E(G))$, or $f(E(G))$ $\subseteq f(V(G))$, or the vertex
color set $f(V(G))$ is an odd-set and the edge color set $f(E(G))$ is an even-
set;

(v) (ve-matching) there exists a constant $k\,^{\prime\prime}$ such that each
edge $uv$ matches with one vertex $w$ such that
$f(uv)+f(w)=k\,^{\prime\prime}$, and each vertex $z$ matches with one edge
$xy$ such that $f(z)+f(xy)=k\,^{\prime\prime}$, except the _singularity_
$f(x_{0})=\lfloor\frac{p+q+1}{2}\rfloor$;

(vi) (set-ordered) $\max f(X)<\min f(Y)$ (resp. $\min f(X)>\max f(Y)$) for the
bipartition $(X,Y)$ of $V(G)$.

We refer to $f$ as an _edge-difference 6C-labeling_ of $G$. $\square$

Since “set-ordered” is in an edge-difference 6C-labeling, so we have a new
$(k,d)$-total coloring as follows:

######  Definition 30.

∗ If a bipartite $(p,q)$-graph $G$ admits an edge-difference 6C-labeling
$f:V(G)\cup E(G)\rightarrow[1,p+q]$, which induces a Topcode-matrix
$T_{code}(G,f)$, then the graph $G$ admits an _edge-difference
$(k,d)$-6C-labeling_

${\begin{split}&F:X\rightarrow S_{m,0,0,d}=\\{0,d,\dots,md\\}\\\ &F:Y\cup
E(G)\rightarrow S_{p+q,k,0,d}=\\{k,k+d,\dots,k+(p+q)d\\}\end{split}}$ (131)

where $V(G)=X\cup Y$ with $X\cap Y=\emptyset$, and the $(k,d)$-6C-labeling $F$
induces an _edge-difference $(k,d)$-6C Topcode-matrix_ of $G$ as follows

$P_{ara}(G,F)=k\cdot I\,^{0}+d\cdot T_{code}(G,f)$ (132)

where $I\,^{0}$ is the unite Topcode-matrix defined in Eq.(124), and the
Topcode-matrices $I\,^{0}$, $T_{code}(G,f)$ and $P_{ara}(G,F)$ have the same
order $3\times q$ (see Fig.42). $\square$

Figure 42:  A bipartite $(13,12)$-graph $G$ admits an edge-difference
6C-labeling and an edge-difference $(k,d)$-6C-labeling.

######  Remark 9.

Since there are the (edge-difference) odd-6C-labeling and the odd-even
separable (edge-difference) 6C-labeling introduced in [36], so we have the
$(k,d)$-odd-6C Topcode-matrix and the odd-even separable $(k,d)$-6C Topcode-
matrix similarly with that defined in Definition 30. $\parallel$

Motivated from Definition 30, we present the following general definition
about the $W$-constraint $(k,d)$-colorings/labelings:

######  Definition 31.

∗ If a bipartite $(p,q)$-graph $G$ admits a $W$-constraint coloring/labeling
$f$, which induces a Topcode-matrix $T_{code}(G,f)$, then the graph $G$ admits
a _$W$ -constraint $(k,d)$-coloring/labeling_

${\begin{split}&F:X\rightarrow S_{m,0,0,d}=\\{0,d,\dots,md\\}\\\ &F:Y\cup
E(G)\rightarrow S_{n,k,0,d}=\\{k,k+d,\dots,k+nd\\}\end{split}}$ (133)

where $V(G)=X\cup Y$ with $X\cap Y=\emptyset$, and this coloring/labeling $F$
induces a _$(k,d)$ -$W$-constraint Topcode-matrix_ of $G$ as follows

$P_{ara}(G,F)=k\cdot I\,^{0}+d\cdot T_{code}(G,f)$ (134)

where $I\,^{0}$ is the unite Topcode-matrix defined in Eq.(124), and
$I\,^{0}$, $T_{code}(G,f)$ and $P_{ara}(G,F)$ are three Topcode-matrices of
$3\times q$. $\square$

######  Theorem 39.

∗ Suppose that a bipartite $(p,q)$-graph $G$ admits different set-ordered
total colorings $f_{1},f_{2},\dots,f_{m}$, and these total colorings are
equivalent from each other. Then the graph $G$ admits $W$-constraint
$(k,d)$-colorings $F_{1},F_{2},\dots,F_{m}$ defined in Definition 31, such
that these $(k,d)$-$W$-constraint Topcode-matrices

$P_{ara}(G,F_{i})=k\cdot I\,^{0}+d\cdot T_{code}(G,f_{i}),~{}i\in[1,m]$ (135)

are equivalent from each other, where three Topcode-matrices $I\,^{0}$,
$T_{code}(G,f_{i})$ and $P_{ara}(G,F_{i})$ have the same order $3\times q$.

###  6.2 Number-based strings containing parameters

Number-based strings containing parameterized numbers are called _$(k,d)$
-type number-based strings_.

######  Example 11.

There are nine Hanzi-graphs $H_{1},H_{2},\dots,H_{9}$ shown in Fig.(43)with
their parameterized Topcode-matrices

$P_{ara}(H_{i},F_{i})=k_{i}\cdot I\,^{0}+d_{i}\cdot
T_{code}(H_{i},g_{i}),~{}i\in[1,9]$ (136)

where the Topcode-matrices $I\,^{0}$, $T_{code}(H_{i},h_{i})$ and
$P_{ara}(H_{i},F_{i})$ for $i\in[1,9]$ have the same order.

We let the Hanzi-graph $H_{1}$ to be a _public-key_ and the Hanzi-graph
$H_{2}$ to be a _private-key_ , and then the Hanzi-graph $H_{3}$ is just the
_topological authentication_ of two Hanzi-graphs $H_{1}$ and $H_{2}$. On the
other hands, the Hanzi-graph $H_{1}$ has its own private-keys $H_{7}$ and
$H_{8}$, and the Hanzi-graph $H_{2}$ has its own private-keys
$H_{4},H_{5},H_{6}$ and $H_{9}$.

In Eq.(138), we can get $(k,d)$-type number-based strings with arbitrary
length of bytes from two Topcode-matrices $P_{ara}(H_{1},F_{1})$ and
$P_{ara}(H_{2},F_{2})$ for the values of $k_{1},d_{1},k_{2}$ and $d_{2}$ with
random bytes. As all of $k_{1}$, $k_{2}$, $d_{1}$ and $d_{2}$ are prime
numbers, moreover, we can ask for

(1) larger odd integer $m=k_{1}\cdot k_{2}$ and larger odd integer
$n=d_{1}\cdot d_{2}$;

(2) larger odd integer $m=k_{1}\cdot k_{2}$ and larger even integer
$n=d_{1}+d_{2}$;

(3) larger even integer $r=k_{1}+k_{2}$ and larger odd integer $s=d_{1}\cdot
d_{2}$;

(4) larger even integer $r=k_{1}+k_{2}$ and larger even integer
$s=d_{1}+d_{2}$.
This is mainly to increase the cost of attackers as much as possible.
$\square$

$T_{code}(H_{1},g_{1})=\left(\begin{array}[]{ccccccccccccccc}3&3&3&3&2&0&0&0&0\\\
1&2&3&4&5&6&7&8&9\\\
4&5&6&7&7&6&7&8&9\end{array}\right),T_{code}(H_{2},g_{2})=\left(\begin{array}[]{ccccccccccccccc}2&1&1&1&1&0&0\\\
1&2&3&4&5&6&7\\\ 3&3&4&5&6&6&7\end{array}\right)$ (137)

$\centering{\begin{split}&\quad P_{ara}(H_{1},F_{1})\\\
&=\left(\begin{array}[]{ccccccccccccccc}3d_{1}&3d_{1}&3d_{1}&3d_{1}&2d_{1}&0&0&0&0\\\
k_{1}+d_{1}&k_{1}+2d_{1}&k_{1}+3d_{1}&k_{1}+4d_{1}&k_{1}+5d_{1}&k_{1}+6d_{1}&k_{1}+7d_{1}&k_{1}+8d_{1}&k_{1}+9d_{1}\\\
k_{1}+4d_{1}&k_{1}+5d_{1}&k_{1}+6d_{1}&k_{1}+7d_{1}&k_{1}+7d_{1}&k_{1}+6d_{1}&k_{1}+7d_{1}&k_{1}+8d_{1}&k_{1}+9d_{1}\end{array}\right)\\\
&\quad
P_{ara}(H_{2},F_{2})=\left(\begin{array}[]{ccccccccccccccc}2d_{2}&d_{2}&d_{2}&d_{2}&d_{2}&0&0\\\
k_{2}+d_{2}&k_{2}+2d_{2}&k_{2}+3d_{2}&k_{2}+4d_{2}&k_{2}+5d_{2}&k_{2}+6d_{2}&k_{2}+7d_{2}\\\
k_{2}+3d_{2}&k_{2}+3d_{2}&k_{2}+4d_{2}&k_{2}+5d_{2}&k_{2}+6d_{2}&k_{2}+6d_{2}&k_{2}+7d_{2}\end{array}\right)\end{split}}\@add@centering$
(138)

Figure 43: The Hanzi-graphs $H_{1},H_{2},H_{5}$ and $H_{6}$ admit set-ordered
graceful labelings, and other Hanzi-graphs admit flawed set-ordered graceful
labelings.

######  Problem 14.

We present a problem for the parametric reconstitution of number-based
strings, called PRONBS-problem as follows:

> PRONBS-problem. For a given number-based string $s=c_{1}c_{2}\cdots c_{n}$
> with $c_{i}\in[0,9]$, rewrite this number-based string $s$ into $3q$
> segments $a_{1},a_{2},\dots,a_{3q}$ with $a_{j}=b_{j,1}b_{j,2}\cdots
> b_{j,m_{j}}$ for $j\in[1,3q]$, such that each number $c_{i}$ of the number-
> based string $s$ is in a segment $a_{j}$ but $c_{i}$ is not in other segment
> $a_{s}$ if $s\neq j$. Find two integers $k_{0},d_{0}\geq 0$, such that each
> segment $a_{j}$ with $j\in[1,3q]$ can be expressed as
> $a_{j}=\beta_{j}k_{0}+\gamma_{j}d_{0}$ for integers
> $\beta_{j},\gamma_{j}\geq 0$ and $j\in[1,3q]$; and moreover find a colored
> $(p,q)$-graph $G$ admitting a parameterized coloring $F$ holding its own
> parameterized Topcode-matrix
>
> $P_{ara}(G,F)_{3\times q}=k\cdot I\,^{0}_{3\times q}+d\cdot
> T_{code}(G,f)_{3\times q}$ (139)
>
> where $f$ is a $W$-constraint coloring of $G$, as well as the parameterized
> number-based string $a_{1}a_{2}\dots
> a_{3q}=c\,^{\prime}_{1}c\,^{\prime}_{2}\cdots c\,^{\prime}_{n}$ is just
> generated from the parameterized Topcode-matrix $P_{ara}(G,F)_{3\times q}$
> when $(k,d)=(k_{0},d_{0})$.

######  Example 12.

When $(k_{2},d_{2})=(7,5)$ in $P_{ara}(H_{2},F_{2})$ of Eq.(138), we get a
Topcode-matrix

$\centering{\begin{split}P_{ara}(H_{2},F_{2})&=\left(\begin{array}[]{ccccccccccccccc}2\cdot
5&5&5&5&5&0&0\\\ 7+5&7+2\cdot 5&7+3\cdot 5&7+4\cdot 5&7+5\cdot 5&7+6\cdot
5&7+7\cdot 5\\\ 7+3\cdot 5&7+3\cdot 5&7+4\cdot 5&7+5\cdot 5&7+6\cdot
5&7+6\cdot 5&7+7\cdot 5\end{array}\right)\\\
&=\left(\begin{array}[]{ccccccccccccccc}10&5&5&5&5&0&0\\\
12&17&22&27&32&37&42\\\ 22&22&27&32&37&37&42\end{array}\right)_{3\times
7}\end{split}}\@add@centering$ (140)

which induces the following two number-based strings

${\begin{split}D_{1}&=105555001217222732374222222732373742\\\
D_{2}&=121722273237420022222732373742105555\end{split}}$

And we can rewrite the $(7,5)$-type number-based string $D_{1}$ into 21
segments:

$0\cdot 7+2\cdot 5$, $0\cdot 7+1\cdot 5$, $0\cdot 7+1\cdot 5$, $0\cdot
7+1\cdot 5$, $0\cdot 7+1\cdot 5$, $0\cdot 7+0\cdot 5$, $0\cdot 7+0\cdot 5$,

$1\cdot 7+1\cdot 5$, $1\cdot 7+2\cdot 5$, $1\cdot 7+3\cdot 5$, $1\cdot
7+4\cdot 5$, $1\cdot 7+5\cdot 5$, $1\cdot 7+6\cdot 5$, $1\cdot 7+7\cdot 5$,

$1\cdot 7+3\cdot 5$, $1\cdot 7+3\cdot 5$, $1\cdot 7+4\cdot 5$, $1\cdot
7+5\cdot 5$, $1\cdot 7+6\cdot 5$, $1\cdot 7+6\cdot 5$, $1\cdot 7+7\cdot 5$.
The above 21 segments form just the parameterized Topcode-matrix
$P_{ara}(H_{2},F_{2})$ shown in Eq.(140).

By rearranging the orders of numbers in the above $(7,5)$-type number-based
strings $D_{1}$ and $D_{2}$, we rewrite these two $(7,5)$-type number-based
strings as follows

${\begin{split}D\,^{\prime}_{1}&=000111222232342222223233425555777777\\\
D\,^{\prime}_{2}&=111737373737374400055552222222222222\end{split}}$

Clearly, it is quite difficult to recover the $(7,5)$-type number-based
strings $D_{1}$ and $D_{2}$ from these two number-based strings
$D\,^{\prime}_{1}$ and $D\,^{\prime}_{2}$, and it is impossible to do such
works for number-based strings with thousands of bytes. $\square$

######  Definition 32.

∗ Parameterized string-colorings and set-colorings. By Definition 7 and
Definition 8, let $G$ be a bipartite $(p,q)$-graph, and its vertex set
$V(G)=X\cup Y$ with $X\cap Y=\emptyset$ such that each edge $uv\in E(G)$ holds
$u\in X$ and $v\in Y$. There are a group of colorings

${\begin{split}&f_{s}:X\rightarrow S_{m,0,0,d}=\\{0,d,\dots,md\\}\\\
&f_{s}:Y\cup E(G)\rightarrow S_{n,k,0,d}=\\{k,k+d,\dots,k+nd\\}\end{split}}$

here it is allowed $f_{s}(u)=f_{s}(w)$ for some distinct vertices $u,w\in
V(G)$ with $s\in[1,B]$ with integer $B\geq 2$, such that
$f_{s}\in\\{$gracefully $(k_{s},d_{s})$-total coloring, odd-gracefully
$(k_{s},d_{s})$-total coloring, edge anti-magic $(k_{s},d_{s})$-total
coloring, harmonious $(k_{s},d_{s})$-total coloring, odd-elegant
$(k_{s},d_{s})$-total coloring, edge-magic $(k_{s},d_{s})$-total coloring,
edge-difference $(k_{s},d_{s})$-total coloring, felicitous-difference
$(k_{s},d_{s})$-total coloring, graceful-difference $(k_{s},d_{s})$-total
coloring, odd-edge edge-magic $(k_{s},d_{s})$-total coloring, odd-edge edge-
difference $(k_{s},d_{s})$-total coloring, odd-edge felicitous-difference
$(k_{s},d_{s})$-total coloring, odd-edge graceful-difference
$(k_{s},d_{s})$-total coloring$\\}$ with $s\in[1,B]$. Then we have:

(i) The bipartite $(p,q)$-graph $G$ admits a _parameterized total string-
coloring_ $F$ defined by

$F(u)=f_{i_{1}}(u)f_{i_{2}}(u)\cdots
f_{i_{B}}(u),F(uv)=f_{j_{1}}(uv)f_{j_{2}}(uv)\cdots
f_{j_{B}}(uv),F(v)=f_{s_{1}}(v)f_{s_{2}}(v)\cdots f_{s_{B}}(v)$

true for each edge $uv\in E(G)$, where $f_{i_{1}}(u)f_{i_{2}}(u)\cdots
f_{i_{B}}(u)$ is a permutation of $f_{1}(u),f_{2}(u),\cdots$, and $f_{B}(u)$,
$f_{j_{1}}(uv)f_{j_{2}}(uv)\cdots f_{j_{B}}(uv)$ is a permutation of
$f_{1}(uv),f_{2}(uv),\cdots,f_{B}(uv)$, as well as $f_{s_{1}}(v)f_{s_{2}}(v)$
$\cdots f_{s_{B}}(v)$ is a permutation of $f_{1}(v),f_{2}(v),\cdots,f_{B}(v)$.

(ii) The bipartite $(p,q)$-graph $G$ admits a _parameterized total set-
coloring_ $\theta$ defined by

${\begin{split}\theta(u)&=\\{f_{1}(u),f_{2}(u),\cdots,f_{B}(u)\\},\quad\theta(uv)=\\{f_{1}(uv),f_{2}(uv),\cdots,f_{B}(uv)\\},\\\
\theta(v)&=\\{f_{1}(v),f_{2}(v),\cdots,f_{B}(v)\\}\end{split}}$

true for each edge $uv\in E(G)$. $\square$

######  Problem 15.

Let $s_{\infty}=c_{1}c_{2}\dots$ be an infinite number-based string with
$c_{i}\in[0,9]$. Does this infinite string $s_{\infty}$ contain any group of
finite number-based strings $S_{1}$, $S_{2}$, $\dots$, $S_{n}$, where
$S_{j}=a_{j,1}a_{j,2}\cdots a_{j,r_{j}}$ with $a_{j,t}\in[0,9]$ for
$t\in[1,r_{j}]$, such that these finite number-based strings form an algebraic
group, or a Topcode-matrix?

###  6.3 Graphic lattices related with parameterized total colorings

####  6.3.1 Operation-type graphic lattices

######  Definition 33.

∗ Suppose that a _graph base_
$\textbf{{H}}=(H_{1},H_{2},\dots,H_{m})=(H_{k})^{m}_{k=1}$ is made by disjoint
bipartite and connected graphs $H_{1}$, $H_{2}$, $\dots$, $H_{m}$, and we
define flawed $W$-constraint $(k,d)$-total colorings for H as follows: Add the
edges of an edge set $E^{*}$ to the base H for joining disjoint bipartite and
connected graphs $H_{1},H_{2},\dots,H_{m}$ to be a connected graph, denoted as

$G=\textbf{{H}}+E^{*}=E^{*}[\ominus]^{m}_{k=1}H_{k}$

If $G=\textbf{{H}}+E^{*}$ admits a $W$-constraint $(k,d)$-total coloring $f$,
then this coloring $f$ induces a $(k,d)$-total coloring $f\,^{\prime}$ of the
base H, we call $f\,^{\prime}$ a _flawed $W$-constraint $(k,d)$-total
coloring_, and we call “$[\ominus]$” to be _edge-join operation_. $\square$

For a graph base $\textbf{{H}}=(H_{1},H_{2},\dots,H_{m})=(H_{k})^{m}_{k=1}$
and $a_{k}\in Z^{0}$, the notation $a_{k}H_{k}$ stands for $a_{k}$ copies of a
graph $H_{k}$, and we use the edges of some edge set $E^{*}$ to join graphs
$a_{1}H_{1}$, $a_{2}H_{2}$, $\dots$, $a_{m}H_{m}$ together, such that the
resultant graph is a connected graph and denoted as
$E^{*}[\ominus]^{m}_{k=1}a_{k}H_{k}$, notice that the edge set $E^{*}$ is
various, and there are more methods to get
$E^{*}[\ominus]^{m}_{k=1}a_{k}H_{k}$, then we get an _edge-join graphic
lattice_

$\textbf{{L}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{H}})=\big{\\{}E^{*}[\ominus]^{m}_{k=1}a_{k}H_{k}:~{}a_{k}\in
Z^{0},H_{k}\in\textbf{{H}},E^{*}\in\textbf{{E}}\big{\\}}$ (141)

with $\sum^{m}_{k=1}a_{k}\geq 1$, and call H _lattice base_ , where E is the
set of edge sets.

Let $G_{i_{1}},G_{i_{2}},\dots,G_{i_{M}}$ be a permutation of graphs
$a_{1}H_{1}$, $a_{2}H_{2}$, $\dots$, $a_{m}H_{m}$ for
$M=\sum^{m}_{k=1}a_{k}\geq 1$. By the vertex-coinciding operation “$[\odot]$”,
we vertex-coincide a vertex $x$ of $G_{i_{j}}$ with a vertex $y$ of
$G_{i_{j+1}}$ into one vertex $x\odot y$ for $j\in[1,M-1]$, the resultant
graph is denoted as

$G_{i_{1}}[\odot]G_{i_{2}}[\odot]\dots[\odot]G_{i_{M}}=[\odot_{\textrm{linear}}]^{m}_{k=1}a_{k}H_{k}$

Hence, we get at least $M!$ graphs
$[\odot_{\textrm{linear}}]^{m}_{k=1}a_{k}H_{k}$. We call the following set

$\textbf{{L}}([\odot_{\textrm{linear}}]Z^{0}\textbf{{H}})=\big{\\{}[\odot_{\textrm{linear}}]^{m}_{k=1}a_{k}H_{k}:~{}a_{k}\in
Z^{0},H_{k}\in\textbf{{H}}\big{\\}}$ (142)

a _linear vertex-coinciding graphic lattice_ with $\sum^{m}_{k=1}b_{k}\geq 1$.

In general, we use the vertex-coinciding operation to graphs
$a_{1}H_{1},a_{2}H_{2},\dots,a_{m}H_{m}$ for producing graphs
$[\odot]^{m}_{k=1}a_{k}H_{k}$, and get a _vertex-coinciding graphic lattice_

$\textbf{{L}}([\odot]Z^{0}\textbf{{H}})=\big{\\{}[\odot]^{m}_{k=1}\beta_{k}H_{k}:~{}\beta_{k}\in
Z^{0},H_{k}\in\textbf{{H}}\big{\\}}$ (143)

with $\sum^{m}_{k=1}\beta_{k}\geq 1$.

Similarly, we do the edge-join operation “$[\ominus]$” and the vertex-
coinciding operation “$[\odot]$” to graphs
$a_{1}H_{1},a_{2}H_{2},\dots,a_{m}H_{m}$ by the mixed way, the resultant graph
is denoted as $[\ominus\odot]^{m}_{k=1}a_{k}H_{k}$ simply. Thereby, we get an
_operation-mixed graphic lattice_

$\textbf{{L}}([\ominus\odot]Z^{0}\textbf{{H}})=\big{\\{}[\ominus\odot]^{m}_{k=1}c_{k}H_{k}:~{}a_{k}\in
Z^{0},H_{k}\in\textbf{{H}}\big{\\}}$ (144)

with $\sum^{m}_{k=1}c_{k}\geq 1$.

It is important to investigate the coloring closure of graphic lattices
$\textbf{{L}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{H}})$,
$\textbf{{L}}([\odot]Z^{0}\textbf{{H}})$ and
$\textbf{{L}}([\ominus\odot]Z^{0}\textbf{{H}})$.

######  Theorem 40.

Since any graph $G$ can be vertex-split into pairwise disjoint graphs
$G_{1},G_{2},\dots,G_{m}$, which is just a permutation of graphs $a_{1}H_{1}$,
$a_{2}H_{2}$, $\dots$, $a_{m}H_{m}$, then each graph is in a vertex-coinciding
graphic lattice $\textbf{{L}}([\odot]Z^{0}\textbf{{H}})$ defined in Eq.(143).

######  Problem 16.

Does there exist another graph base
$\textbf{{H}}\,^{\prime}=(H\,^{\prime}_{1},H\,^{\prime}_{2},\dots,H\,^{\prime}_{m})$
holding one of

$\textbf{{L}}([\ominus]Z^{0}\textbf{{H}}\,^{\prime})=\textbf{{L}}([\ominus]Z^{0}\textbf{{H}}),~{}\textbf{{L}}([\odot]Z^{0}\textbf{{H}}\,^{\prime})=\textbf{{L}}([\odot]Z^{0}\textbf{{H}}),~{}\textbf{{L}}([\ominus\odot]Z^{0}\textbf{{H}}\,^{\prime})=\textbf{{L}}([\ominus\odot]Z^{0}\textbf{{H}})$

A _parameter-colored tree base_
$\textbf{{H}}^{c}=(H^{c}_{1},H^{c}_{2},\dots,H^{c}_{m})$ consists of disjoint
graphs $H^{c}_{1},H^{c}_{2}$, $\dots$, $H^{c}_{m}$, where each tree
$H^{c}_{i}$ admits a $W_{i}$-constraint $(k_{i},d_{i})$-total coloring $g_{i}$
($i\in[1,m]$) defined in Definition 7, and two vertex color sets
$g_{i}(V(H^{c}_{i}))\cap g_{j}(V(H^{c}_{j}))\neq\emptyset$. We take is a
permutation $G_{1},G_{2},\dots,G_{Q}$ of graphs
$c_{1}H^{c}_{1},c_{2}H^{c}_{2},\dots,c_{n}H^{c}_{m}$ with
$Q=\sum^{m}_{k=1}c_{k}$, and we vertex-coincide a vertex $x$ of $H^{c}_{i}$
with a vertex $y$ of $H^{c}_{i+1}$ into one vertex $x\odot y$ for
$i\in[1,M-1]$, where $f_{i}(x)=f_{j}(y)$ according to the assume, so the
resultant graph is a tree, denoted as

$G_{1}[\odot]G_{2}[\odot]\cdots[\odot]G_{Q}=[\odot_{\textrm{linear}}]^{m}_{k=1}c_{k}H^{c}_{k}$

and we can get $Q!$ trees for each number $Q=\sum^{m}_{k=1}c_{k}$. We define a
_linear parameter-colored graphic lattice_ as follows

$\textbf{{L}}([\odot_{\textrm{linear}}]Z^{0}\textbf{{H}}^{c})=\big{\\{}[\odot_{\textrm{linear}}]^{m}_{k=1}c_{k}H^{c}_{k}:~{}c_{k}\in
Z^{0},H^{c}_{k}\in\textbf{{H}}^{c}\big{\\}}$ (145)

with $\sum^{m}_{k=1}c_{k}\geq 1$.

In general, we do the vertex-coinciding operation to disjoint graphs
$H^{c}_{1},H^{c}_{2},\dots,H^{c}_{m}$, the resultant simple graphs are denoted
as $[\odot]^{m}_{k=1}\gamma_{k}H^{c}_{k}$, and we define a _parameter-colored
graphic lattice_ as follows

$\textbf{{L}}([\odot]Z^{0}\textbf{{H}}^{c})=\big{\\{}[\odot]^{m}_{k=1}\gamma_{k}H^{c}_{k}:~{}\gamma_{k}\in
Z^{0},H^{c}_{k}\in\textbf{{H}}^{c}\big{\\}}$ (146)

with $\sum^{m}_{k=1}\gamma_{k}\geq 1$.

####  6.3.2 Uncolored tree-graphic lattices

An _uncolored tree base_
$\textbf{{T}}=(T_{1},T_{2},\dots,T_{n})=(T_{k})^{n}_{k=1}$ consists of
disjoint trees $T_{1},T_{2},\dots,T_{n}$. We use the edges of an edge set
$E^{*}$ to join these disjoint trees together, the resulting graph is denoted
as $\textbf{{T}}+E^{*}$. Notice that one tree $H_{i}$ may be used two or more
times in $\textbf{{T}}+E^{*}$, so we write
$E^{*}[\ominus]^{n}_{k=1}\alpha_{k}T_{k}$ with $\sum^{m}_{k=1}\alpha_{k}\geq
1$ for replacing $\textbf{{T}}+E^{*}$ by the _edge-join operation_
“$[\ominus]$”. We have a _tree-base graphic lattice_ as follows:

$\textbf{{L}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})=\big{\\{}E^{*}[\ominus]^{m}_{k=1}\mu_{k}T_{k}:~{}\mu_{k}\in
Z^{0},T_{k}\in\textbf{{T}},E^{*}\in\textbf{{E}}\big{\\}}$ (147)

with $\sum^{m}_{k=1}\mu_{k}\geq 1$. And moreover, we have other two tree-base
graphic lattices

$\textbf{{L}}([\odot]Z^{0}\textbf{{T}})=\big{\\{}[\odot]^{m}_{k=1}\lambda_{k}T_{k}:~{}\lambda_{k}\in
Z^{0},T_{k}\in\textbf{{T}}\big{\\}}$ (148)

with $\sum^{m}_{k=1}\lambda_{k}\geq 1$ and

$\textbf{{L}}([\ominus\odot]Z^{0}\textbf{{T}})=\big{\\{}[\ominus\odot]^{m}_{k=1}\gamma_{k}T_{k}:~{}\gamma_{k}\in
Z^{0},T_{k}\in\textbf{{T}}\big{\\}}$ (149)

with $\sum^{m}_{k=1}\gamma_{k}\geq 1$.

For considering the coloring closure of graphic lattices, let
$\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$ be the
set of all trees in a tree-base graphic lattice
$\textbf{{L}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$ defined in Eq.(147),
and we call it _all-tree-graphic lattice_. Similarly, we have other two all-
tree-graphic lattices
$\textbf{{L}}_{\textrm{tree}}([\odot]Z^{0}\textbf{{T}})\subset\textbf{{L}}([\odot]Z^{0}\textbf{{T}})$
defined in Eq.(148) and
$\textbf{{L}}_{\textrm{tree}}([\ominus\odot]Z^{0}\textbf{{T}})\subset\textbf{{L}}([\ominus\odot]Z^{0}\textbf{{T}})$
defined in Eq.(149). By the theorems introduced in the pre-subsections, these
three all-tree-graphic lattices are coloring closure for the $(k,d)$-total
colorings stated in the following theorem 41:

######  Theorem 41.

Each tree contained in the all-tree-graphic lattices
$\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$,
$\textbf{{L}}_{\textrm{tree}}([\odot]Z^{0}\textbf{{T}})$ and
$\textbf{{L}}_{\textrm{tree}}([\ominus\odot]Z^{0}\textbf{{T}})$ admits the
following ten $(k,d)$-total colorings: graceful $(k,d)$-total coloring,
harmonious $(k,d)$-total coloring, (odd-edge) edge-difference $(k,d)$-total
coloring, (odd-edge) graceful-difference $(k,d)$-total coloring, (odd-edge)
felicitous-difference $(k,d)$-total coloring, (odd-edge) edge-magic
$(k,d)$-total coloring.

####  6.3.3 $(k,d)$-colored tree-graphic lattices

A _parameter-colored tree base_
$\textbf{{T}}^{c}=(T^{c}_{1},T^{c}_{2},\dots,T^{c}_{n})$ is consisted of
disjoint trees $T^{c}_{1},T^{c}_{2}$, $\dots$, $T^{c}_{n}$, where each tree
$T^{c}_{i}$ admits a $W_{i}$-constraint $(k_{i},d_{i})$-total coloring $f_{i}$
($i\in[1,n]$) defined in Definition 7. If $W=W_{1}=W_{2}=\cdots=W_{n}$, and
$(k,d)=(k_{1},d_{1})=(k_{2},d_{2})=\cdots=(k_{n},d_{n})$, we say that
$\textbf{{T}}^{c}$ is a _uniformly $W$-constraint $(k,d)$-colored tree base_.

Assume that two vertex color sets $f_{i}(V(T^{c}_{i}))\cap
f_{j}(V(T^{c}_{j}))\neq\emptyset$ for $i\neq j$ in $\textbf{{T}}^{c}$. We take
is a permutation $I_{1},I_{2},\dots,I_{M}$ of graphs
$b_{1}T^{c}_{1},b_{2}T^{c}_{2},\dots,b_{n}T^{c}_{n}$ with
$M=\sum^{n}_{k=1}b_{k}$, and we vertex-coincide a vertex $x$ of $T^{c}_{i}$
with a vertex $y$ of $T^{c}_{i+1}$ into one vertex $x\odot y$ for
$i\in[1,M-1]$, where $f_{i}(x)=f_{j}(y)$ according to the assume, so the
resultant graph is a tree, denoted as

$I_{1}[\odot]I_{2}[\odot]\cdots[\odot]I_{M}=[\odot_{\textrm{linear}}]^{n}_{k=1}b_{k}T^{c}_{k}$

And moreover we get a _linear parameter-colored all-tree-graphic lattice_

$\textbf{{L}}_{\textrm{tree}}([\odot_{\textrm{linear}}]Z^{0}\textbf{{T}}^{c})=\big{\\{}[\odot_{\textrm{linear}}]^{n}_{k=1}b_{k}T^{c}_{k}:~{}b_{k}\in
Z^{0},T^{c}_{k}\in\textbf{{T}}^{c}\big{\\}}$ (150)

with $\sum^{n}_{k=1}b_{k}\geq 1$.

In general, we call the following set

$\textbf{{L}}_{\textrm{tree}}([\odot]Z^{0}\textbf{{T}}^{c})=\big{\\{}[\odot]^{n}_{k=1}\lambda_{k}T^{c}_{k}:~{}\lambda_{k}\in
Z^{0},T^{c}_{k}\in\textbf{{T}}^{c}\big{\\}}$ (151)

a _parameter-colored all-tree-graphic lattice_ with
$\sum^{n}_{k=1}\lambda_{k}\geq 1$.

######  Remark 10.

Sometimes, a tree
$H\in\textbf{{L}}_{\textrm{tree}}([\odot]Z^{0}\textbf{{T}}^{c})$ may be
colored by two or more different $W_{i}$-constraint $(k_{i},d_{i})$-total
colorings defined in Definition 7. Thereby, the Topcode-matrix $T_{code}(H)$
contains tow or more groups of parameters
$(k_{i},d_{i}),(k_{j},d_{j}),\cdots,(k_{t},d_{t})$. This enables us to create
very complex number-base springs. $\parallel$

For a permutation $I_{1},I_{2},\dots,I_{M}$ of graphs
$\beta_{1}T^{c}_{1},\beta_{2}T^{c}_{2},\dots,\beta_{n}T^{c}_{n}$ with
$M=\sum^{n}_{k=1}\beta_{k}$, we add leaves to each tree $I_{j}$ and get a
$W_{i}$-constraint $(k_{i},d_{i})$-total coloring $f^{*}_{j}$ based on the
$W_{i}$-constraint $(k_{i},d_{i})$-total coloring admitted by the parameter-
colored tree base $\textbf{{T}}^{c}$, the resultant tree is denoted as
$I^{*}_{j}$, and two vertex color sets $f^{*}_{i}(V(I^{*}_{i}))\cap
f^{*}_{j}(I^{*}_{j}))\neq\emptyset$ by the assume $f_{i}(V(T^{c}_{i}))\cap
f_{j}(V(T^{c}_{j}))\neq\emptyset$ for $i\neq j$ in $\textbf{{T}}^{c}$. And
then we do the vertex-coincide operation to
$I^{*}_{1},I^{*}_{2},\dots,I^{*}_{M}$, that is

$I^{*}_{1}[\odot]I^{*}_{2}[\odot]\cdots[\odot]I^{*}_{M}=[\ominus_{\textrm{leaf}}\odot_{\textrm{linear}}]^{n}_{k=1}\beta_{k}T^{c}_{k}$

and get a _parameter-colored adding-leaf tree-graphic lattice_

$\textbf{{L}}_{\textrm{tree}}([\ominus_{\textrm{leaf}}\odot_{\textrm{linear}}]Z^{0}\textbf{{T}}^{c})=\big{\\{}[\ominus_{\textrm{leaf}}\odot_{\textrm{linear}}]^{n}_{k=1}\beta_{k}T^{c}_{k}:~{}\beta_{k}\in
Z^{0},T^{c}_{k}\in\textbf{{T}}^{c}\big{\\}}$ (152)

with $\sum^{n}_{k=1}\beta_{k}\geq 1$.

######  Remark 11.

Sometimes, a tree
$H\in\textbf{{L}}_{\textrm{tree}}([\odot]Z^{0}\textbf{{T}}^{c})$, or
$H\in\textbf{{L}}_{\textrm{tree}}([\ominus_{\textrm{leaf}}\odot_{\textrm{linear}}]Z^{0}\textbf{{T}}^{c})$
may be colored by two or more different $W_{i}$-constraint
$(k_{i},d_{i})$-total colorings defined in Definition 7. Thereby, the Topcode-
matrix $T_{code}(H)$ contains tow or more groups of parameters
$(k_{i},d_{i}),(k_{j},d_{j}),\cdots,(k_{t},d_{t})$. This enables us to create
very complex number-base springs. $\parallel$

We have some mathematical problems as follows:

######  Problem 17.

About an all-tree-graphic lattice
$\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$ defined
in Eq.(147), we may consider the following problems:

  1. Ext-1.

Find a tree
$H^{*}\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$
has its own diameter holding $D(H^{*})\leq D(H)$ for each tree
$H\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$.

  2. Ext-2.

Find a tree
$H^{*}\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$
has its own maximum degree holding $\Delta(H^{*})\leq\Delta(H)$ for each tree
$H\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$.

  3. Ext-3.

Find a subset
$S_{\varepsilon}\subset\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$,
which contains all $\varepsilon$ in
$\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$ for
$\varepsilon\in\\{$caterpillars, lobsters, spiders$\\}$.

  4. Ext-4.

Find a tree
$H^{*}\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$
has its own leaf number $n_{1}(H^{*})$ holding $n_{1}(H^{*})\leq n_{1}(H)$ for
the leaf number $n_{1}(H)$ of each tree
$H\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$.

  5. Ext-5.

Estimate the cardinality of an all-tree-graphic lattice
$\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$.

  6. Ext-6.

Find connection between a leaf number $l_{e}(G)$ and the leaf numbers
$l_{e}(T_{i})$ in the uncolored tree base T for
$G\in\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$.

  7. Ext-7.

For the subset
$S_{\textrm{matching}}\subset\textbf{{L}}_{\textrm{tree}}(\textbf{{E}}[\ominus]Z^{0}\textbf{{T}})$,
in which each tree has a perfect matching. Does each tree $J\in
S_{\textrm{matching}}$ admits one of strongly $W$-constraint $(k,d)$-total
colorings?

######  Problem 18.

Consider problems shown in Problem 17 about other two all-tree-graphic
lattices $\textbf{{L}}_{\textrm{tree}}([\odot]Z^{0}\textbf{{T}})$ and
$\textbf{{L}}_{\textrm{tree}}([\ominus\odot]Z^{0}\textbf{{T}})$.

######  Problem 19.

In general, we have a colored all-tree-graphic lattice
$\textbf{{L}}_{\textrm{tree}}([\bullet]Z^{0}\textbf{{T}})$ based on a graph
operation “$[\bullet]$” and each tree
$T\in\textbf{{L}}_{\textrm{tree}}([\bullet]Z^{0}\textbf{{T}})$ admits a
$W$-constraint $(k,d)$-total coloring, where “$[\bullet]$” is one of
“$[\ominus]$”, “$[\odot]$” and “$[\ominus\odot]$”. It may be good to consider
the following problems:

  1. Prob-1.

Estimate the cardinality of an all-tree-graphic lattice
$\textbf{{L}}_{\textrm{tree}}([\bullet]Z^{0}\textbf{{T}})$.

  2. Prob-2.

Find a tree $H^{*}\in\textbf{{L}}_{\textrm{tree}}([\bullet]Z^{0}\textbf{{T}})$
admitting a $W$-constraint $(k,d)$-total coloring $f$ having its own number
$|f(V(H^{*}))|$ of different colors colored to the vertices of the tree
$H^{*}$ under the graceful $(k,d)$-total coloring $f$ to hold
$|f(V(H^{*}))|\leq|h(V(H))|$ for each tree
$H\in\textbf{{L}}_{\textrm{tree}}([\bullet]Z^{0}\textbf{{T}})$ admitting a
$W$-constraint $(k,d)$-total coloring $h$.

  3. Prob-3.

Same lattice base. For two given connected graphs $G_{i}$ and $G_{j}$, if
there exists a base $H=(H_{k})^{m}_{k=1}$ and a graph operation “$[\bullet]$”,
such that $G_{i}=[\bullet]^{m}_{k=1}a_{k}H_{k}$ and
$G_{j}=[\bullet]^{m}_{k=1}b_{k}H_{k}$, we say that both connected graphs
$G_{i}$ and $G_{j}$ obey the _same lattice base_. Here, we are interesting on
$|V(H_{k})|\geq 2$, or some $H_{i}$ is not a star, that is $H_{i}\neq
K_{1,s}$.

  4. Prob-4.

If both connected graphs $G_{i}$ and $G_{j}$ contain a proper subgraph
$G^{*}$, we contract $G^{*}$ into a vertex $w^{*}$, so we get two connected
graphs $G_{i}\odot G^{*}$ and $G_{j}\odot G^{*}$, such that $G_{i}=(G_{i}\odot
G^{*})\odot^{-1}w^{*}$ and $G_{j}=(G_{j}\odot G^{*})\odot^{-1}w^{*}$. If both
connected graphs $G_{i}\odot G^{*}$ and $G_{j}\odot G^{*}$ obey the same
lattice base for any proper subgraph $G^{*}$ holding $G^{*}\subset G_{i}$ and
$G^{*}\subset G_{j}$, can we claim both connected graphs $G_{i}$ and $G_{j}$
obey the same lattice base?

####  6.3.4 Tree-graphic lattice homomorphisms

######  Definition 34.

[2] A _graph homomorphism_ $G\rightarrow H$ from a graph $G$ into another
graph $H$ is a mapping $f:V(G)\rightarrow V(H)$ such that $f(u)f(v)\in E(H)$
for each edge $uv\in E(G)$. $\square$

Totally-colored graph homomorphisms. The authors in [37] propose a new type of
graph homomorphisms by combining graph homomorphisms and graph total colorings
together as follows:

######  Definition 35.

[37] Let $G\rightarrow H$ be a graph homomorphism from a $(p,q)$-graph $G$ to
another $(p\,^{\prime},q\,^{\prime})$-graph $H$ based on a mapping
$\alpha:V(G)\rightarrow V(H)$ such that $\alpha(u)\alpha(v)\in E(H)$ for each
edge $uv\in E(G)$. The graph $G$ admits a total coloring $f$, the graph $H$
admits a total coloring $g$. Write the edge color set $f(E(G))=\\{f(uv):uv\in
E(G)\\}$ and the edge color set
$g(E(H))=\\{g(\alpha(u)\alpha(v)):\alpha(u)\alpha(v)\in E(H)\\}$, there are
the following conditions:

  1. (C-1)

$V(G)=X\cup Y$, each edge $uv\in E(G)$ holds $u\in X$ and $v\in Y$ true.
$V(H)=W\cup Z$, each edge $\alpha(u)\alpha(v)\in E(G)$ holds $\alpha(u)\in W$
and $\alpha(v)\in Z$ true.

  2. (C-2)

$f(uv)=|f(u)-f(v)|$ for each edge $uv\in E(G)$,
$g(\alpha(u)\alpha(v))=|g(\alpha(u))-g(\alpha(v))|$ for each
$\alpha(u)\alpha(v)\in E(H)$.

  3. (C-3)

$f(uv)=g(\alpha(u)\alpha(v))$ for each edge $uv\in E(G)$.

  4. (C-4)

$f(x)\in[1,q+1]$ for $x\in V(G)$, and $g(y)\in[1,q\,^{\prime}+1]$ for $y\in
V(H)$.

  5. (C-5)

$f(x)\in[1,2q+2]$ for $x\in V(G)$, and $g(y)\in[1,2q\,^{\prime}+2]$ for $y\in
V(H)$.