Eℓ-sets and Eℓ-1-sets of star $\ell$-set transposition graphs

Italo J. Dejter

University of Puerto Rico

Rio Piedras, PR 00936-8377

<EMAIL_ADDRESS>
###### Abstract

Let $0<\ell\in\mathbb{Z}$. The notion of an efficient dominating set of a
graph $G$, also said to be a perfect code of $G$, is generalized to that of an
efficient dominating$\,{}^{\ell}$-set (Eℓ-set), or perfectℓcode and applied to
the cases of vertex transitive star $\ell$-set transposition graphs
($j=\ell,\ell-1$), i.e. based on permutations of finite strings with each
element repeated $j$ times, with applications to total coloring, error-
correcting codes and networks.

##  1 Introduction

Let $0<\ell\in\mathbb{Z}$. Given a finite graph $G=(V(G),E(G))$ of girth
larger than 3 and a subset $S\subseteq V(G)$, we say that $S$ is an efficient
dominating$\,{}^{\ell}\\!$-set (Eℓ-set) or a perfect$\,{}^{\ell}\\!$code, if
for each $v\in V(G)\setminus S$ there are exactly $\ell$ vertices
$v^{0},v^{1},\ldots,v^{\ell-1}$ in $S$ such that $v$ is adjacent to $v^{i}$,
for every $i\in[\ell]=\\{0,\ldots,\ell-1\\}$. Since the girth of $G$ is larger
than 3, then the subset $S(v)=\\{v^{0},v^{1},\ldots,v^{\ell-1}\\}$ of $V(G)$
is an independent set of $G$ said to be a dominating $\ell$-set (D$\ell$S) of
$v$ with respect to $S$, written wrt $S$. In particular:

  1. (a)

if $\ell=1$, then $S$ is an efficient dominating set (E-set) [1, 2, 3, 6, 7,
11], also called a 1-perfect code [4, 5]; in this case, $S$ provides a perfect
packing of $G$ by balls of radius 1, also said to be 1-spheres; if $G$ is
$r$-regular, the sphere packing condition $|V(G)|=(r+1)|S|$ is a necessary
condition for $S$ to be an E-set of $G$ [6];

  2. (b)

if $\ell>1$, then $v$ is the only vertex of $G$ in the intersection of the
1-spheres of $v^{0},v^{1},\ldots,v^{\ell-1}$; we deal with this extension case
of an E-set from Section 2 on.

In Section 2 we recall the definition of the family of vertex transitive star
$\ell$-set transposition graphs $G=ST^{\ell}_{k}$, ($1<k\in\mathbb{Z}$), which
are a particular case of the graphs treated in [10], in such a case in a
context of determining Gray codes and Hamilton paths and cycles.

In Sections 3 and 4, we present Eℓ-sets and Eℓ-1-sets of the graphs
$ST^{\ell}_{k}$, respectively, adapting the setting of E-chains in [6] with
applications of total colorings of graphs, error-correcting codes and networks
(see also Remark 9).

##  2 Star $\ell$-set transposition graphs

For $0<\ell\in\mathbb{Z}$ and $2\leq k\in\mathbb{Z}$, we say that a string
over the alphabet $[k]=\\{0,\ldots,k-1\\}$ that contains exactly $\ell$
occurrences of $i$, for each $i\in[k]$, is an $\ell$-set permutation. In
denoting specific $\ell$-set permutations, commas and brackets will be usually
omitted. Let $V^{\ell}_{k}$ be the set of all $\ell$-set permutations of
length $k\ell$.

Let $ST^{\ell}_{k}$ be the graph on vertex set $V^{\ell}_{k}$ with an edge
between any two vertices $v=v_{0}v_{1}\cdots v_{k\ell-1}$ and
$w=w_{0}w_{1}\cdots w_{k\ell-1}$ differing in a star $\ell$-set transposition,
i.e. obtained by swapping the first entry $v_{0}$ of $v=v_{0}v_{1}\cdots
v_{k\ell-1}\in V^{\ell}_{k}$ with some entry $v_{j}$
($j\in[k\ell]\setminus\\{0\\}$) whose value differs from that of $v_{0}$ (so
$v_{j}\neq v_{0}$), thus obtaining either $w=w_{0}\cdots w_{j}\cdots
w_{k\ell-1}=v_{j}\cdots v_{0}\cdots w_{k\ell-1}$ or $w=w_{0}\cdots
w_{k\ell-1}=v_{k\ell-1}\cdots v_{0}$. Note that $ST^{\ell}_{k}$ has
$\frac{(k\ell)!}{(\ell!)^{k}}$ vertices and regular degree $(k-1)\ell$.

It is known that all $k$-permutations, (that is all 1-set permutations of
length $k$), form the symmetric group, denoted $Sym_{k}$, under composition of
$k$-permutations, each $k$-permutation $v_{0}v_{1}\cdots v_{k-1}$ taken as a
bijection from the identity $k$-permutation $01\cdots(k-1)$ onto
$v_{0}v_{1}\cdot v_{k-1}$ itself. A graph $ST^{1}_{k}$ with $k>1$ (which
excludes $ST^{1}_{1}$) is the Cayley graph of $Sym_{k}$ with respect to the
set of transpositions $\\{(0\;i);i\in[k]\setminus\\{0\\}\\}$. Such a graph
$ST_{k}^{1}$ is denoted $ST_{k}$ in [1, 6], where is proven that its vertex
set admits a partition into $k$ E-sets, exemplified on the upper left of
Figure 1 for $ST^{1}_{3}=ST_{3}$, with the vertex parts of the partition
differentially colored in black, red and green, for respective first entries
0, 1 and 2. Figure 1 of [6] shows a similar example for $ST^{1}_{4}=ST_{4}$.
Note that the graphs $ST^{\ell}_{k}$ are vertex transitive, but they are
neither Cayley or Shreier graphs for $\ell>1$.

##  3 Eℓ-sets of the graphs $ST^{\ell}_{k}$

The vertices of $ST^{\ell}_{k}$ are the $\ell$-set permutations $v_{0}\dots
v_{k\ell-1}$ of the string

$\overbrace{0\cdots 0}\overbrace{1\cdots 1}\overbrace{2\cdots
2}\cdots\overbrace{(k-1)\cdots(k-1)}=0^{\ell}1^{\ell}2^{\ell}\cdots(k-1)^{\ell}.$

Let us see, for each $i\in[k]$ with $ST^{\ell}_{k}\neq ST^{1}_{2}$, that the
vertices $v=v_{0}\dots v_{k\ell-1}$ of $ST^{\ell}_{k}$ with first entry
$v_{0}$ equal to $i$ form an Eℓ-set $S=S_{i}^{k}$ of $ST^{\ell}_{k}$. Since
$ST^{\ell}_{k}\neq ST^{1}_{2}$ has girth larger than 3, then each D$\ell$S wrt
$S_{i}^{k}$ of $ST^{\ell}_{k}$ is an independent set of $ST^{\ell}_{k}$.

Figure 1: The 6-cycles $ST^{1}_{3}=ST_{3}$ and $ST^{2}_{2}$, and the Desargues
graph $ST^{3}_{2}$.

######  Theorem 1.

Let $0<\ell,k\in\mathbb{Z}$ and let $ST^{\ell}_{k}\neq ST^{1}_{2}$. For each
$i\in[k]$, the $\ell$-set permutations $v_{0}\dots v_{k\ell-1}$ of
$0^{\ell}1^{\ell}2^{\ell}\cdots(k-1)^{\ell}$ with first entry $v_{0}$ equal to
$i$ form an E$\,{}^{\ell}$-set $S_{i}^{k}$ of $ST^{\ell}_{k}$.

###### Proof.

For fixed $i\in[k]$, each vertex $v=v_{0}v_{1}\cdots,v_{k\ell-1}$ of
$E(ST^{\ell}_{k})\setminus S_{i}^{k}$ has initial entry $v_{0}=j$, for some
$j\in[k]$ such that $j\neq i$. Then, $v$ is adjacent to $\ell$ vertices of
$S_{i}^{k}$ obtained by transposing the position of each of the $\ell$ entries
$v_{h}=i$, ($h\in\\{1,\ldots,k\ell-1\\}$), with the position of that initial
entry $v_{0}=j$. The graph induced by the edges of such adjacencies form a
copy $H$ of the complete bipartite graph $K_{1,\ell}$ (a ball of radius 1)
with $v$ as its sole degree-$\ell$ vertex (the center of the ball) and its
leaves (if $H$ is taken as a rooted tree with $v$ as its root) as the vertices
$v^{j}$ in $S_{i}^{k}$ obtained from $v$ by transposing $v_{0}=i$ with those
$v_{h}=j$. ∎

######  Corollary 2.

The vertex set $V(ST^{\ell}_{k})$ admits a partition into $k$
E$\;{}^{\ell}$-sets $S_{i}^{k}$, where $i\in[k]$.

###### Proof.

The Eℓ-sets $S_{i}^{k}$ form a partition of $V(ST^{\ell}_{k})$, since each
such $S_{i}^{k}$ is composed precisely by the vertices $v=v_{0}v_{1}\cdots
v_{k\ell-1}$ of $ST^{\ell}_{k}$ having initial entry $v_{0}=i$, which form one
of the $k$ parts of the partition. ∎

Let $2ST^{\ell}_{k}$ be the multigraph obtained from $ST^{\ell}_{k}$ by
replacing each edge $e$ of $ST^{\ell}_{k}$ by two parallel edges with the same
endvertices of $e$.

######  Corollary 3.

Let $i\in[k]$. Each vertex of $S_{i}^{k}$ in $ST^{\ell}_{k}$ belongs to
$k\ell-1$ D$\,\ell$Ss wrt $S_{i}^{k}$, where the induced graph of each such
D$\,\ell$S is isomorphic to the complete bipartite graph $K_{1,\ell}$. The set
of all such D$\ell$Ss, $\forall i\in[k]$, forms a partition of the edge set of
$2ST^{\ell}_{k}$.

###### Proof.

Since there are $k-1$ values $j\neq i$ in $[k]$, each vertex $v\in S_{i}^{k}$
belongs to $k\ell-1$ D$\ell$Ss wrt $S_{i}^{k}$. Since each such $v$ is the
neighbor of $\ell$ vertices with first entry $v_{0}=j$, for each $j\in[k]$
with $j\neq i$, then the induced graph of each such D$\ell$S is isomorphic to
$K_{1,\ell}$ , noted already in the proof of Theorem 1. Also, each edge $e$ of
$ST^{\ell}_{k}$ with endvertices $v$ and $w$ having respective first entries
$i$ and $j$ is both a member of $S_{i}^{k}$ and of $S_{j}^{k}$. Thus, $e$
appears in $2ST^{\ell}_{k}$ as two parallel edges with endvertices in
$S_{i}^{k}$ and $S_{j}^{k}$. ∎

######  Example 4.

The graph $ST^{2}_{2}$ is the 6-cycle graph $(0011,1001,0101,1100,0110,1010)$,
represented in the middle left of Figure 1, showing in distinct edge shades
the induced subgraphs of the composing D$2$Ss of the E2-set
$S_{0}^{2}=\\{0011,0101,0110\\}$, namely the D$2$S $\\{0011,0101\\}$ of 1001
wrt $S_{0}^{2}$, the D$2S\,${0011,0110} of 1010 wrt $S_{0}^{2}$, and the D$2$S
$\\{0110,0101\\}$ of 1100 wrt $S_{0}^{2}$.

######  Example 5.

The graph $ST^{3}_{2}$ is the Desargues graph drawn on the right of Figure 1,
where each subgraph $K_{1,3}$ induced by a D$3$Ss $S_{0}^{2}(v)$ of a vertex
$v=0v_{1}\cdots v_{5}$ of $ST^{3}_{2}$ wrt $S_{0}^{2}$ has its edges in three
pairwise different colors. In fact, the edge colors in this representation of
$ST^{3}_{2}$ are seen to define ten monochromatic copies of $K_{1,3}$ centered
at the vertices of the form $w=1w_{1}\cdots w_{5}$ in colors red, blue, green,
hazel and violet (two vertex-disjoint monochromatic copies of $K_{1,3}$ per
color), illustrating that each $v\in S_{0}^{2}$ is the intersection of
$\ell=3$ such balls, each ball contributing just one edge incident to $v$,
where the three resulting edge colors are pairwise different. Those
monochromatic subgraphs $K_{1,3}$ appear in “opposing” pairs, allowing to
raise the question of how many colors are necessary to color such edge
partitions.

######  Example 6.

The vertices of $ST^{2}_{3}$ are the $\ell$-set permutations, ($\ell=2$), of
$v_{0}^{0}=001122$, yielding a total of $\frac{6!}{2!2!2!}=\frac{720}{8}=90$
vertices. The regular degree of $ST^{2}_{3}$ is 4. The graph $ST^{2}_{3}$ has
the E3-set $S_{0}^{2}=\\{v_{0}\cdots v_{5}\in V(ST^{2}_{3});v_{0}=0\\}$. For
example, $v=100122$ has $S^{k}_{0}(v)=S^{2}_{0}(v)=\\{010122,001122\\}$ as its
D$\ell$S wrt $S$, and $v^{\prime}=120120$ has
$S^{k}_{0}(v^{\prime})=S^{2}_{0}(v^{\prime})=\\{021120,020121\\}$ as D$\ell$S
wrt $S_{0}^{2}$. Each vertex $v$ in $S_{0}^{2}$ belongs to $k\ell-1=4$ D$2$Ss
wrt $S_{0}^{2}$. While $ST^{2}_{3}$ has 90 vertices, $S_{0}^{2}$ has
$\frac{90}{k}=\frac{90}{3}=30$ vertices. For example, 010122 belongs to
$S^{2}_{0}(100122)$, $S^{2}_{0}(110022)$, $S^{2}_{0}(210102)$ and
$S^{3}_{0}(210120)$. Specifically, as in display (5):

$\displaystyle\begin{array}[]{ccc}S_{0}^{3}(100122)=\\{010122,001122\\},\\\
S_{0}^{3}(110022)=\\{010122,011022\\},\\\
S_{0}^{3}(210102)=\\{010122,012102\\},\\\
S_{0}^{3}(210120)=\\{010122,012120\\}.\end{array}$ (5)

######  Example 7.

The vertices of $ST^{3}_{3}$ are the $3$-set permutations of
$v_{0}^{0}=000111222$, yielding a total of
$|V(ST^{3}_{3})|=\frac{(k\ell)!}{\ell!^{k}}=\frac{9!}{3!^{3}}=1680$ vertices.
The regular degree of $ST^{3}_{3}$ is $k\ell-1=3\times 2=6$. The graph
$ST^{3}_{3}$ has the set of 9-tuples $S_{0}^{3}=\\{v_{0}\cdots v_{8}\in
V(ST^{3}_{3});a_{0}=0\\}$ as an E3-set. For example, $100011222$ has
$\\{000111222,001011222,010011222\\}$ as its D$3$S wrt $S_{0}^{3}$, and
$120120120$ has $\\{021120120,020121120,020120121\\}$ as its D$3$S wrt
$S_{0}^{3}$. While $ST^{3}_{3}$ has $1680$ vertices, $S_{0}^{3}$ has
$\frac{1680}{k}=\frac{1680}{3}=560$ vertices. Each vertex of $S_{0}^{3}$
belongs to $k\ell-1=6$ D$3$Ss wrt $S_{0}^{3}$. For example, 010011222 belongs
to the sets in display (12).

$\displaystyle\begin{array}[]{c}S_{0}^{3}(100011222)=\\{010011222,001011222,000111222\\},\\\
S_{0}^{3}(110001222)=\\{010011222,010101222,011001222\\},\\\
S_{0}^{3}(110010222)=\\{010011222,010011222,010001222\\},\\\
S_{0}^{3}(210011022)=\\{010011222,010211022,012001022\\},\\\
S_{0}^{3}(210011202)=\\{010011222,010211202,012001202\\},\\\
S_{0}^{3}(210011220)=\\{010011222,010211220,012001200\\}.\end{array}$ (12)

######  Corollary 8.

Let $i\in[k]$ and let $S_{i}^{k}$ be an E$\,{}^{\ell}$-set of $ST^{\ell}_{k}$.
For each fixed vertex $v=v_{0}v_{1}\cdots v_{k\ell-1}\in
V(ST^{\ell}_{k})\setminus S_{i}^{k}$, the $\ell$ vertices of the $D\ell$S
$S_{i}^{k}(v)$ (wrt $S_{i}^{k}$) bear bijectively the $\ell$ occurrences of
$i$, each such occurrence as a value of a corresponding non-initial entry
$v_{h}$ of $v$, ($h\in\\{1,\ldots,k\ell-1\\}$), transposed with the value $j$
at its initial entry.

###### Proof.

The behavior described in the statement is exemplified in Examples 6-7
(respective displays (5)-(12)), according to the specifications. It is
likewise for larger values of $\ell$ and/or $k$. ∎

######  Remark 9.

If $ST^{\ell}_{k}$ is taken as the plan map of a city with streets represented
by edges and corners represented by vertices, then an Eℓ-set $S_{i}^{k}$ may
be planned to hold cops stationed at its vertices. In the case of an event at
a vertex $v$ of $ST^{\ell}_{k}$, if the vertex is in $S_{i}^{k}$, then a
corresponding cop is at hand. Otherwise, there are $\ell$ cops at the vertices
in $S_{i}^{k}(v)$, any of which can be present by moving along one sole edge.
As another application, an error-correcting model of $ST^{\ell}_{k}$ will give
for each received message a total of 1 (in $S_{i}^{k}$) or $\ell$ (in
$ST^{\ell}_{k}\setminus S_{i}^{k}$) corrected messages.

##  4 Eℓ-1-sets of the graphs $ST^{\ell}_{k}$

Let $1<\ell\in\mathbb{Z}$, let
$i\in[k\ell]\setminus\\{0\\}=\\{1,\ldots,k\ell-1\\}$, let $\Sigma_{i}^{k}$ be
the set of vertices $v_{0}v_{1}\cdots v_{k\ell-1}$ of $ST^{\ell}_{k}$ such
that $v_{0}=v_{i}$, ($i=1,\ldots,k\ell-1$), and let $E_{i}^{k}$ be the set of
edges having color $i$ in $G\setminus\Sigma_{i}^{k}$. We will show that
$\Sigma_{i}^{k}$ is an E-set of $ST^{\ell}_{k}$. Clearly, no edge of
$E_{i}^{k}$ is incident to the vertiices of $\Sigma_{i}^{k}$.

We recall that a total coloring of a graph $G$ is an assignment of colors to
the vertices and edges of $G$ such that no two incident or adjacent elements
(vertices or edges) are assigned the same color [9]. A total coloring of $G$
such that the vertices adjacent to each $v\in V(G)$ together with $v$ itself
are assigned pairwise different colors will be said to be an efficient
coloring. The efficient coloring will be said to be totally efficient if $G$
is $k$-regular, the color set is $[k]=\\{0,1,\ldots,k-1\\}$ and each $v\in
V(G)$ together with its neighbors are assigned all the colors in $[k]$. The
total (resp. efficient) chromatic number $\chi^{\prime\prime}(G)$ (resp.
$\chi^{\prime\prime\prime}(G)$) of $G$ is defined as the least number of
colors required by a total (resp. efficient) coloring of $G$.

######  Theorem 10.

(I) Let $k>1$, let $i\in[k\ell]\setminus\\{0\\}=\\{1,\ldots,k\ell-1\\}$ and
let $\Sigma_{i}^{k}$ be the set of vertices $v_{0}v_{1}\dots v_{k\ell-1}$ of
$ST^{\ell}_{k}$ such that $v_{0}=v_{i}$. Then, $V^{\ell}_{k}$ admits a vertex
partition into $k\ell-1$ E-sets $\Sigma_{i}^{k}$,
($i\in[k\ell]\setminus\\{0\\}$). (II) Let $k>2$, let
$j\in[k\ell]\setminus\\{0\\}$ and let $E_{j}^{k}$ be the set of all edges of
color $j$. Then, $ST^{\ell}_{k}\setminus\Sigma_{i}^{k}\setminus E_{i}^{k}$ is
the disjoint union of $k\ell^{k-1}$ copies of $ST^{\ell}_{k-1}$. (III) If
$\ell=2$, then the objects presented in items (I)-(II) form a totally
efficient coloring of $ST^{\ell}_{k}$.

###### Proof.

Item (I): Recall that $ST^{\ell}_{k}$ has $\frac{(k\ell)!}{(\ell!)^{k}}$
vertices and regular degree $(k-1)\ell$. Let $i=k\ell-1$ and let
$j\in[k\ell]$. Then, each vertex $v=v_{0}v_{1}\cdots
v_{k\ell-3}v_{k\ell-2}v_{k\ell-1}=0v_{1}\cdots v_{k\ell-3}j0$ is the neighbor
of vertex $w=jv_{1}\cdots v_{k\ell-3}00$ via an edge of color $k-1$. Item
(II): $v\in\Sigma_{i}^{k}=\Sigma_{k\ell-1}^{k}$. Being $w$ at distance 1 from
$\Sigma_{k\ell-1}^{k}$, then $w$ is in the open neighborhood
$N(\Sigma_{i}^{k})$ [6] of $\Sigma_{k\ell-1}^{k}$ in $ST^{\ell}_{k}$, so $w\in
N(\Sigma_{i}^{k})=N(\Sigma_{k\ell-1}^{k})\subseteq
ST^{\ell}_{k}\setminus\Sigma_{i}^{k}\setminus
E_{i}^{k}=ST^{\ell}_{k}\setminus\Sigma_{k\ell-1}^{k}\setminus
E_{k\ell-1}^{k}$. In fact, $N(\Sigma_{i}^{k})=N(\Sigma_{k\ell-1}^{k})$ is a
connected component of $ST^{\ell}_{k}\setminus\Sigma_{i}^{k}\setminus
E_{i}^{k}=ST^{\ell}_{k}\setminus\Sigma_{k\ell-1}^{k}\setminus
E_{k\ell-1}^{k}$. A similar conclusion holds for each other open neighborhoods
$N(\Sigma_{i}^{k})$, ($1\leq i<k\ell-1$). Item (III): If $\ell=2$, then there
is a sole color not employed in coloring the edges incident to any particular
vertex of $ST^{\ell}_{k}$, providing its totally efficient coloring via the
items (I)-(II). ∎

######  Example 11.

The graph $ST^{2}_{2}$ of Example 4 has also the totally efficient coloring
depicted on the lower left of Figure 1, where $\Sigma_{1}^{2}=\\{0011,1100\\}$
is color blue, as is $E_{1}^{2}=\\{(0101,1001),(0110,1010)\\}$;
$\Sigma_{2}^{2}=\\{0101,1010\\}$ is color green, as is
$E_{2}^{2}=\\{(0110,1100),$ $(0011,1001)\\}$; $\Sigma_{3}^{2}=\\{0110,1001\\}$
is color red, as is $E_{3}^{2}=\\{(0011,1010),(0101,1100)\\}$.

######  Example 12.

The Desargues graph $ST^{3}_{2}$ of Example 5 will now also be shown to
contain the E2-set $\Sigma_{5}^{2}$ depicted in Figure 2, (in contrast to the
E3-set depicted on the right of Figure 1), and formed by the vertices having
the first, ($v_{0}$), and the last, ($v_{5}$), entries with a common value,
(either 0 or 1). In fact, each subgraph $K_{1,3}$ induced by a D$3$Ss
$\Sigma_{5}^{2}(v)$ of a vertex $v=v_{0}v_{1}\cdots v_{4}v_{5}=iv_{1}\cdots
v_{4}j$ of $ST^{3}_{2}$ wrt $\Sigma_{5}^{2}$, ($i\neq j$), has its edges other
than its sole dashed black-colored edge in a maximum of two pairwise different
colors. Moreover, the edge colors in this representation of $ST^{3}_{2}$ are
seen to define eight monochromatic copies of $K_{1,3}$ centered at the
vertices of the form $w=w_{0}w_{1}\cdots w_{4}w_{5}=jw_{1}\cdots w_{4}j$ in
thick colors red, hazel, green and blue (two vertex-disjoint monochromatic
copies of $K_{1,3}$ per color), illustrating that each $v\in
ST^{3}_{2}\setminus\Sigma_{5}^{2}$ is the intersection of $\ell-1=2$ such
balls, each ball contributing just one edge incident to $v$, where the two
resulting edge colors are distinct. Those monochromatic subgraphs $K_{1,3}$
appear in “opposing” pairs, allowing to raise the question of how many colors
are necessary to color such edge partitions. However, observe that: (a) the
1-factor $E_{5}^{2}$ conformed by the dashed black-colored edges induces
$V(ST^{3}_{2})\setminus\Sigma_{5}^{2}$ and that (b) we can assign the said
thick edge colors red, hazel, green and blue to the numbers $j=1$, 2, 3 and 4,
respectively, so that the sole thick edge incident to any leaf vertex of a
monochromatic copy of $K_{1,3}$ outside such a copy of $K_{1,3}$ has color $j$
if and only if $j$ is the color number of such $K_{1,3}$.

Figure 2: The Desargues graph $ST^{3}_{2}$ revisited.

######  Remark 13.

The total coloring of $ST^{2}_{k}$ will be referred to as its color structure.
The $k2^{k-1}$ copies of $ST^{2}_{k-1}$ in $ST^{2}_{k}$ whose disjoint union
is $ST^{2}_{k}\setminus\Sigma_{i}^{k}\setminus E_{i}^{k}$ inherit each a color
structure that generalizes that of the 3-colored 6-cycles in
$ST^{2}_{3}\setminus\Sigma_{5}^{3}$ and is similar to the color structure of
$ST^{2}_{k-1}$.

######  Example 14.

The graph $ST^{2}_{3}$ has the E-set $\Sigma_{5}^{3}$ with 18 vertices denoted
as in display (16):

$\displaystyle\begin{array}[]{cccccc}A=011220,&\underline{A}=022110,&B=012210,&\underline{B}=021120,&C=012120,&\underline{C}=021210,\\\
D=122001,&\underline{D}=100221,&E=120021,&\underline{E}=102201,&F=120201,&\underline{F}=102021,\\\
G=200112,&\underline{G}=211002,&H=201102,&\underline{H}=210012,&J=201012,&\underline{J}=210102.\\\
\end{array}$ (16)

This example is further expanded in [8].

######  Corollary 15.

Let $k>2$. Then:

  1. 1.

$ST^{2}_{k}$ has $\frac{2k!}{2^{k}}$ vertices having $\frac{2k!}{2^{k}(2k-1)}$
vertices in each color $1,2,\ldots,2k-1$;

  2. 2.

$ST^{2}_{k}$ has $\frac{2k!}{2^{k}}\times(k-1)$ edges;

  3. 3.

color $k\ell-1$ provides exactly $\frac{2k!}{2^{k}(2k-1)}=y$ vertices forming
a PDS $\Sigma_{2k-1}^{k}$ of $ST^{2}_{k}$;

  4. 4.

the $y$ resulting dominating copies of $K_{1,2k-2}$ have a total of
$y\times(2k-2)=z$ edges;

  5. 5.

there are exactly $\frac{2k!}{2^{k}}\times(k-1)-z=h$ edges in $ST_{2k-1}^{k}$
not counted in item 4;

  6. 6.

the $h$ edges in item 5. contain $\frac{h}{2k-1}$ edges in each color
$1,2,\ldots,2k-1$;

  7. 7.

so they contain $h-\frac{h}{2k-1}$ edges in colors $\neq 2k-1$, (namely,
$1,2,\ldots,2k-2$);

  8. 8.

there are $\frac{2k!}{2^{k}}-y$ vertices in
$ST^{2}_{k}\setminus\Sigma_{2k-1}^{k}$ dominated by $\Sigma_{2k-1}^{k}$;

  9. 9.

the $\frac{2k!}{2^{k}}-y$ vertices in item 8. appear in $k\times(2k-2)$ copies
of $ST^{2}_{k-1}$;

  10. 10.

there are $\frac{h}{(2k-1)^{2}k}$ edges in each copy of $ST_{2k-1}^{k}$ in
$ST^{2}_{k}\setminus\Sigma_{2k-1}^{k}$.

###### Proof.

The ten items of the corollary can be verified directly from the enumerative
facts involved with the graphs $ST^{2}_{k}$. ∎

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