$w_{3}z_{3}$, and $w_{5}z_{4}$. Let $H$ be the e-graph with the vertex set
$S\cup\\{a,b,c\\}$, edges $z_{\pi(1)}a$, $z_{\pi(2)}a$, $ab$, $bc$,
$cz_{\pi(3)}$, and $cz_{\pi(4)}$, and with $d_{H}(a)=d_{H}(b)=d_{H}(c)=3$, see
Figure 32. Note that for any $11/4$-coloring $\varphi$ of $H$, Observation
2.10 implies $|(\varphi(z_{\pi(1)})\cup\varphi(z_{\pi(2)}))\cap\varphi(c)|\leq
3$ and $|(\varphi(z_{\pi(3)})\cup\varphi(z_{\pi(4)}))\cap\varphi(c)|=0$, and
thus
$\bigl{|}\varphi(c)\setminus\bigl{(}\bigcup_{i=1}^{4}\varphi(z_{i})\bigr{)}\bigr{|}\geq
1$, and $\bigl{|}\bigcup_{i=1}^{4}\varphi(z_{i})\bigr{|}\leq 10$. The standard
argument for the configuration $G_{1}$ attaching at $S$ and the replacement
graph $H$ enforcing $\bigl{|}\bigcup_{i=1}^{4}\varphi(z_{i})\bigr{|}\leq 10$
gives an e-graph $G^{\prime}$ and its proper induced sub-e-graph
$F\in\mathcal{C}_{0}$ (note that in the computer-assisted argument to exclude
$F=G^{\prime}$, we can assume that $z_{\pi(1)}\neq z_{\pi(2)}$ and
$z_{\pi(3)}\neq z_{\pi(4)}$, since otherwise $H$ would contain two nailed
vertices and by (a0), $F$ would be $C_{5}$ or $K_{4}^{+}$; and since the
nailed vertices of $F$ would be adjacent, $F$ would also contain two adjacent
non-nailed vertices of degree two, implying that $G$ contains two adjacent
vertices of degree two in contradiction to Lemma 6.6).

By Corollary 6.8, the graph $G^{\prime}$ is $2$-edge-connected, implying by
Observation 2.6 that $F$ contains at least two nailed vertices distinct from
the nailed vertex $b$ of $G^{\prime\prime}$. By (a0), it follows that
$b\not\in V(F)$ and the underlying graph of $F$ is $C_{5}$ or $K_{4}^{+}$.
Since $G$ is critical, $F$ is not an induced sub-e-graph of $G$, and thus we
can assume $a\in V(F)$, $a$ is a nailed vertex of $F$, and $z_{\pi(1)}\neq
z_{\pi(2)}$. Since both $z_{\pi(1)}$ and $z_{\pi(2)}$ have degree three in
$G^{\prime\prime}$ and $F$ has only one nailed vertex other than $a$, $F$ is
not a 5-cycle. If the underlying graph of $F$ is $K_{4}^{+}$, then
$z_{\pi(1)}$ or $z_{\pi(2)}$ is the other nailed vertex of $F$ and $F$
contains two adjacent non-nailed vertices of degree two. These vertices have
degree two in $G$ as well, contradicting Lemma 6.6. ∎

######  Lemma 8.4.

Suppose $C_{1}=v_{1}v_{2}v_{3}v_{4}v_{5}$ and
$C_{2}=w_{1}w_{2}w_{3}w_{4}w_{5}$ are disjoint cycles in a smallest
counterexample $G$ and $v_{1}w_{1},v_{3}w_{3}\in E(G)$. If $\deg v_{2}=2$,
then $\deg w_{2}=3$.

###### Proof.

Suppose for a contradiction that $\deg w_{2}=2$. Note that
$v_{4}w_{4},w_{5}v_{5}\not\in E(G)$ by Lemma 7.3. If $v_{4}w_{5}\in E(G)$,
then Corollary 6.8 implies that either $v_{5}w_{4}\in E(G)$, or $v_{5}$ and
$w_{4}$ have a common neighbor of degree two; however, both such graphs are
$11/4$-colorable. Therefore $v_{5}w_{4}\not\in E(G)$, and symmetrically
$v_{4}w_{5}\not\in E(G)$. Let $z_{1}$, $z_{2}$, $z_{3}$, and $z_{4}$ be the
neighbors of $v_{4}$, $v_{5}$, $w_{4}$, and $w_{5}$, respectively, not
belonging to $C_{1}\cup C_{2}$.

Suppose $i,j\in\\{1,2,3,4\\}$ are distinct, $z_{i}=z_{j}$, and $\deg z_{i}=2$.
Letting $\\{k,l\\}=\\{1,2,3,4\\}\setminus\\{i,j\\}$, Corollary 6.8 implies
that $z_{k}=z_{l}$ also has degree two; but then $G$ is $11/4$-colorable (or
contains a triangle), which is a contradiction. Hence, if $\deg z_{i}=2$, then
$z_{i}$ is distinct from all vertices $z_{j}$ for
$j\in\\{1,2,3,4\\}\setminus\\{i\\}$; we let $z^{\prime}_{i}$ be the neighbor
of $z_{i}$ not in $C_{1}\cup C_{2}$. If $\deg z_{i}=3$, we let
$z^{\prime}_{i}=z_{i}$.

$v_{5}$$v_{1}$$v_{2}$$v_{3}$$v_{4}$$z_{2}$$z_{1}$$w_{5}$$w_{1}$$w_{2}$$w_{3}$$w_{4}$$z_{4}$$z_{3}$
$a_{1}$$a_{2}$$a_{3}$$a_{4}$$z_{1}$$z_{2}$$z_{3}$$z_{4}$

Figure 33: The e-graph $G_{1}$ and the replacement e-graph $H$.

Let $S=\\{z^{\prime}_{1},\ldots,z^{\prime}_{4}\\}$ and let $G_{1}$ be the sub-
e-graph of $G$ consisting of $G[V(C_{1})\cup V(C_{2})]$, the edges
$v_{4}z_{1}$, $v_{5}z_{2}$, $w_{4}z_{3}$, $w_{5}z_{4}$, and the edges
$z_{i}z^{\prime}_{i}$ for $i\in\\{1,2,3,4\\}$ such that $\deg z_{i}=2$, see
Figure 33. If at least one of the vertices $z_{1}$, …, $z_{4}$ has degree two,
then the configuration $G_{1}$ attaching at $S$ is excluded by reducibility.
Hence, suppose that all vertices $z_{1}$, …, $z_{4}$ have degree three. Let
$H$ be the e-graph with the vertex set $S\cup\\{a_{1},a_{2},a_{3},a_{4}\\}$,
edges of the paths $z_{1}a_{1}a_{2}z_{2}$ and $z_{3}a_{3}a_{4}z_{4}$, and with
$d_{H}(a_{i})=3$ for $i\in\\{1,2,3\\}$ and $d_{H}(a_{4})=2$, see Figure 33.
The standard argument for the configuration $G_{1}$ attaching at $S$ and the
replacement graph $H$ enforcing $|\varphi(z_{1})\cap\varphi(z_{2})|\leq 3$ and
$|\varphi(z_{3})\cap\varphi(z_{4})|\leq 2$ gives an e-graph $G^{\prime}$ and
its proper induced sub-e-graph $F\in\mathcal{C}_{0}$ (we have $F\neq
G^{\prime}$ by (a0), since $G^{\prime}$ has three nailed vertices).

Since $G$ is critical, $F$ is not an induced sub-e-graph of $G$ (even after
possibly replacing the path $z_{1}a_{1}a_{2}z_{2}$ by the path
$z_{1}v_{4}v_{5}z_{2}$) and thus $a_{3},a_{4}\in V(F)$. Corollary 6.8 implies
$G^{\prime}$ is connected, and thus by Observation 2.6, $F\neq G^{\prime}$
contains another nailed vertex in addition to $a_{3}$. Hence, by (a0) $F$
contains two nailed vertices and the underlying graph of $F$ is either $C_{5}$
or $K_{4}^{+}$. However, if $F$ were a 5-cycle, then the corresponding
$5$-cycle in $G$ (obtained by replacing the path $z_{3}a_{3}a_{4}z_{4}$ by
$z_{3}w_{4}w_{5}z_{4}$) would only contain three vertices of degree three,
contradicting Corollary 6.7, and the case that the underlying graph of $F$ is
$K_{4}^{+}$ is excluded by Lemma 5.3. ∎

We are now ready to prove the main result of this section.

######  Lemma 8.5.

Every minimum counterexample is $3$-regular.

###### Proof.

Suppose for a contradiction that a minimum counterexample $G$ contains a
vertex $v$ of degree two. By Lemma 8.2 (considering the vertex of degree two
contained in $K$ instead of $v$ if necessary), we can without loss of
generality assume that $v$ is contained in a 5-cycle
$C_{1}=u_{1,1}v_{1}vv_{2}u_{2,1}$. By Corollary 6.7, all vertices of $C_{1}$
except for $v$ have degree three. Let $u_{1,2}$ and $u_{2,2}$ be the neighbors
of $v_{1}$ and $v_{2}$ not in $C_{1}$, respectively; we have $u_{1,2}\neq
u_{2,2}$ and $\deg u_{1,2}=\deg u_{2,2}=3$ by Lemmas 5.1 and Lemma 6.6. By
Lemma 8.2, there exists a $5$-cycle $C_{2}\subset G-\\{v_{1},v_{2},v\\}$
containing another vertex of degree two such that each of $v_{1}$ and $v_{2}$
has exactly one neighbor in $C_{2}$. We claim that these neighbors of $v_{1}$
and $v_{2}$ are non-adjacent.

###### Subproof.

Suppose for a contradiction that the neighbors of $v_{1}$ and $v_{2}$ in
$C_{2}$ are adjacent, and thus we can (by relabelling the vertices if
necessary) assume these neighbors are $u_{1,1}$ and $u_{2,1}$; let
$C_{2}=u_{1,1}u_{2,1}w_{3}w_{4}w_{5}$. By Lemma 8.3, we have $\deg w_{4}=3$,
and thus by symmetry and Corollary 6.7, we can assume $\deg w_{3}=2$ and $\deg
w_{5}=3$.

$u_{1,1}$$v_{1}$$v$$v_{2}$$u_{2,1}$$u_{1,2}$$u_{2,2}$$w_{5}$$w_{4}$$w_{3}$$x$$w^{\prime}_{4}$$x^{\prime}$
Figure 34: The e-graph $G_{1}$.

We claim that $u_{1,2}$ and $w_{5}$ do not have a common neighbor $x$ of
degree $2$; indeed, if they did, let $x^{\prime}$ be the neighbor of $u_{1,2}$
distinct from $v_{1}$ and $x$, let $w^{\prime}_{4}$ be the neighbor of $w_{4}$
not in $C_{2}$, let $S_{1}=\\{x^{\prime},w^{\prime}_{4},u_{2,2}\\}$, and let
$G_{1}$ be the sub-e-graph of $G$ consisting of $G[V(C_{1})\cup V(C_{2})]$,
the path $u_{1,2}xw_{5}$, and the edges $x^{\prime}u_{1,2}$,
$w^{\prime}_{4}w_{4}$, and $u_{2,2}v_{2}$, see Figure 34. The configuration
$G_{1}$ attaching at $S_{1}$ is excluded by reducibility.

Let us now apply Lemma 8.2 again, with $w_{3}$ playing the role of $v$, giving
a $5$-cycle $K\subset G-\\{u_{2,1},w_{3},w_{4}\\}$ containing a vertex of
degree two and a neighbor of $u_{2,1}$ and of $w_{4}$. If $u_{1,1}\in V(K)$,
then $v_{2}\not\in V(K)$ since $u_{2,1}$ has only one neighbor in $K$, and
thus $K$ contains the path $u_{1,2}v_{1}u_{1,1}w_{5}$; however, then $w_{5}$
and $u_{1,2}$ would have a common neighbor of degree two, which is a
contradiction.

Therefore $u_{1,1}\not\in V(K)$, $v_{2}\in V(K)$, and
$K=u_{1,2}v_{1}vv_{2}u_{2,2}$. The vertex $w_{4}$ has a neighbor in $K$. If
$w_{4}u_{2,2}\in E(G)$, then Corollary 6.8 implies that either $u_{1,2}w_{5}$
have a common neighbor of degree two (which we already excluded), or
$u_{1,2}w_{5}\in E(G)$, which contradicts Lemma 7.3. It follows that
$w_{4}u_{1,2}\in E(G)$.

Lemma 7.3 then implies $w_{5}u_{2,2}\not\in E(G)$, as otherwise the vertex
$w_{3}$ of degree two would have a neighbor in the 4-cycle
$w_{4}w_{5}u_{2,2}u_{1,2}$. By Corollary 6.8, it follows that $w_{5}$ and
$u_{2,2}$ have a common neighbor of degree two. However, the corresponding
graph $G$ is $11/4$-colorable, which is a contradiction. ∎

Since the neighbors of $v_{1}$ and $v_{2}$ in $C_{2}$ are non-adjacent, we
cannot have $u_{1,1}\in V(C_{2})$; otherwise, as
$\deg_{G-\\{v_{1},v,v_{2}\\}}u_{1,1}=2$, we would also have $u_{2,1}\in
V(C_{2})$. Symmetrically, $u_{2,1}\not\in V(C_{2})$. Therefore,
$u_{1,2},u_{2,2}\in V(C_{2})$. Let $C_{2}=u_{1,2}w_{1}u_{2,2}w_{2}w_{3}$. By
Lemma 8.4, we have $\deg w_{1}=3$, and thus by symmetry and Corollary 6.7, we
can assume $\deg w_{2}=2$ and $\deg w_{3}=3$.

We claim that $u_{1,1}$ and $w_{1}$ do not have a common neighbor $y$ of
degree $2$: Otherwise Corollary 6.8 and Lemma 6.6 would imply $u_{2,1}w_{3}\in
E(G)$; however, the corresponding graph $G$ is $11/4$-colorable.

Let us now again apply Lemma 8.2, with $w_{2}$ playing the role of $v$, giving
a $5$-cycle $K^{\prime}\subset G-\\{u_{2,2},w_{2},w_{3}\\}$ containing a
vertex of degree two and a neighbor of $u_{2,2}$ and of $w_{3}$. If $w_{1}\in
V(K^{\prime})$, then $v_{2}\not\in V(K^{\prime})$, and thus $K^{\prime}$
contains the path $w_{1}u_{1,2}v_{1}u_{1,1}$; however, then $w_{1}$ and
$u_{1,1}$ would have a common neighbor of degree two, which is a
contradiction.

Therefore, $w_{1}\not\in V(K^{\prime})$ and $v_{2}\in V(K^{\prime})$, which
implies $K^{\prime}=C_{1}$. The vertex $w_{3}$ has a neighbor in $K^{\prime}$,
and $w_{3}$ is not adjacent to $u_{1,1}$ by Lemma 7.3; hence, $w_{3}u_{2,1}\in
E(G)$. However, then Corollary 6.8 and Lemma 7.3 imply $w_{1}$ and $u_{1,1}$
have a common neighbor of degree two, which is a contradiction. ∎

Consequently, we can strengthen Corollary 6.8.

######  Corollary 8.6.

Every minimum counterexample is $3$-edge-connected.

##  9 4-cycles

We now show that a minimum counterexample cannot contain a cycle of length
four.

######  Lemma 9.1.

Every minimum counterexample has girth at least five.

###### Proof.

Suppose for a contradiction $C=v_{1}v_{2}v_{3}v_{4}$ is a 4-cycle in a minimum
counterexample $G$. Note that $C$ is an induced cycle, since $G$ is triangle-
free. For $i\in\\{1,\ldots,4\\}$, let $u_{i}$ be the neighbor of $v_{i}$
outside of $C$. Note that the vertices $u_{1}$, …, $u_{4}$ are pairwise
distinct: otherwise, since $G$ is triangle-free, we could assume
$u_{1}=u_{3}$; but then $v_{2}$ and $v_{4}$ would have the same neighborhood,
contradicting the criticality of $G$. By Lemma 7.3, the vertices $u_{1}$, …,
$u_{4}$ have degree three.

Let $G^{\prime}$ be the e-graph obtained from $G-V(C)$ by setting
$d_{G^{\prime}}(u_{i})=2$ for $i\in\\{1,\ldots,4\\}$. Suppose that
$G^{\prime}$ has an $11/4$-coloring $\varphi^{\prime}$. For
$i\in\\{1,\ldots,4\\}$, let $L_{i}$ be a subset of
$[0,11)\setminus\varphi^{\prime}(u_{i})$ of measure 6. Since the fractional
choosability is equal to the fractional chromatic number [2] and $C$ has
(fractional) chromatic number $2$, there exist sets $A_{i}\subset L_{i}$ for
$i\in\\{1,\ldots,4\\}$ such that $|A_{i}|=3$ and $(A_{1}\cup
A_{3})\cap(A_{2}\cup A_{4})=\emptyset$. For $i\in\\{1,\ldots,4\\}$, let
$M_{i}$ be a subset of $[0,11)\setminus(A_{i-1}\cup A_{i}\cup A_{i+1})$ of
measure $2$, where $A_{0}=A_{4}$ and $A_{5}=A_{1}$. Using the fractional
choosability of $C$ again, for $i\in\\{1,\ldots,4\\}$, there exist sets
$B_{i}\subset M_{i}$ of measure $1$ such that $(B_{1}\cup B_{3})\cap(B_{2}\cup
B_{4})=\emptyset$. Let $\varphi(x)=\varphi^{\prime}(x)$ for any vertex $x$ at
distance at least two from $C$, and for $i\in\\{1,\ldots,4\\}$, let
$\varphi(v_{i})=A_{i}\cup B_{i}$ and let $\varphi(u_{i})$ be a subset of
$\varphi^{\prime}(u_{i})\setminus B_{i}$ of measure $4$. Then $\varphi$ is an
$11/4$-coloring of $G$, which is a contradiction.

Hence, $G^{\prime}$ does not have an $11/4$-coloring and contains a critical
induced sub-e-graph $F$, belonging to $\mathcal{C}_{0}$ by the minimality of
$G$. By a computer-assisted enumeration, we verified that for every graph in
$\mathcal{C}_{0}$ with no nailed vertices and with exactly four vertices of
degree two, attaching a 4-cycle adjacent to these vertices results in an
e-graph that either belongs to $\mathcal{C}_{0}$ or is not critical. Since $G$
arises from $G^{\prime}$ in this way, we have
$G^{\prime}\not\in\mathcal{C}_{0}$, and thus $F\neq G^{\prime}$.

By Corollary 8.6, the graph $G^{\prime}$ is connected, and thus $F$ contains
at least one nailed vertex by Observation 2.6. If $F$ contains exactly one
nailed vertex, then (b0) implies $F$ contains at least three non-nailed
vertices of degree two, and thus $|V(F)\cap\\{u_{1},\ldots,u_{4}\\}|\geq 3$.
However, then $G$ contains at most two edges with exactly one end in $V(F)\cup
V(C)$, contradicting Corollary 8.6.

$v_{1}$$v_{2}$$v_{4}$$v_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$x_{2}$$x_{1}$$x_{1}^{\prime}$$x_{2}^{\prime}$

$v_{1}$$v_{2}$$v_{4}$$v_{3}$$u_{1}$$u_{2}$$u_{3}$$u_{4}$$x_{2}$$x_{1}$$x_{1}^{\prime}$$x_{2}^{\prime}$

Figure 35: Some of the possibilities for the e-graph $G_{1}$.

Therefore, $F$ contains at least two nailed vertices, and by (a0) and Lemma
5.3, $F$ is a 5-cycle with exactly two nailed vertices. Consequently, we can
by symmetry assume $u_{1},u_{2},u_{3}\in V(F)$ and $u_{4}\not\in V(F)$. Let
$x_{1}$ and $x_{2}$ be the nailed vertices of $F$ and let $x^{\prime}_{1}$ and
$x^{\prime}_{2}$ be their neighbors not in $F$. Let
$S=\\{x^{\prime}_{1},x^{\prime}_{2},u_{4}\\}$ and let $G_{1}$ be the sub-e-
graph of $G$ consisting of $G[V(C)\cup V(F)]$ together with the edges
$x_{1}x^{\prime}_{1}$, $x_{2}x^{\prime}_{2}$, and $v_{4}u_{4}$, see Figure 35.
The configuration $G_{1}$ attaching at $S$ is excluded by reducibility. ∎

##  10 3-regular graphs

We now come to the core of the argument, very similar to the one used in [4].
For each vertex $v$ of a minimum counterexample, we delete $v$ and the
neighbors of $v$ and find an $11/4$-coloring of the resulting e-graph. We then
convexly combine these colorings to obtain an $11/4$-coloring of the whole
graph, which gives a contradiction. Of course, an issue is that we need to
argue that no critical subgraph is created in this way, which is generally
straightforward with the exception of 5-cycles with two nailed vertices. To
deal with these 5-cycles, we exploit the fact that we can afford to keep a few
of the vertices at distance two from $v$ nailed. The following definition is
used to determine which vertices to nail.

$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u$$v$

$u_{1}$$u_{2}$$u_{3}$$u_{4}$$u$$v$

$v$$u$$\vec{Z}$

Figure 36: The edges of the difficult 5-cycle graph.

Let $G$ be a minimum counterexample. The _difficult 5-cycle graph_ of $G$ is
defined as the directed graph $\vec{Z}$ with the vertex set $V(G)$, where
$(v,u)\in\vec{Z}$ if and only if $G$ contains a $5$-cycle
$K=uu_{1}u_{2}u_{3}u_{4}$, $v\not\in V(K)$, and $G$ contains three paths
$P_{1}$, $P_{2}$, and $P_{3}$ from $v$ to $K$ of length two, disjoint except
for $v$ and intersecting $K$ only in their last vertices, such that the last
vertices of $P_{1}$, $P_{2}$, and $P_{3}$ are either $\\{u,u_{1},u_{4}\\}$ or
$\\{u,u_{2},u_{3}\\}$, see Figure 36. Let $W(v,K)$ denote the set of last
vertices of $P_{1}$, $P_{2}$, and $P_{3}$ (this set is uniquely defined
without the need to specify $P_{1}$, $P_{2}$, and $P_{3}$, since $G$ has girth
at least five, and thus each neighbor of $v$ can have only one neighbor in
$K$), and let $u(v,K)$ denote the vertex $u$ (again, $u$ is uniquely
determined by $v$ and $K$).

The motivation for the definition of the difficult 5-cycle graph comes from
the following Lemma.

######  Lemma 10.1.

Let $v$ be a vertex of a minimum counterexample $G$ and let $D$ be the set of
outneighbors of $v$ in the difficult $5$-cycle graph $\vec{Z}$ of $G$. Let
$v_{1}$, $v_{2}$, and $v_{3}$ be the neighbors of $v$ in $G$, and let $N$ be
the set of their neighbors distinct from $v$. Let $G_{v}$ be the e-graph
obtained from $G-\\{v,v_{1},v_{2},v_{3}\\}$ by setting $d_{G_{v}}(x)=2$ for
$x\in N\setminus D$ and $d_{G_{v}}(x)=3$ for $x\in D$. Then $G_{v}$ is
$11/4$-colorable.

###### Proof.

Suppose for a contradiction that $G_{v}$ does not have an $11/4$-coloring, and
thus it contains a critical induced sub-e-graph $F$, belonging to
$\mathcal{C}_{0}$ by the minimality of $G$. Note that $|N|=6$ by Lemma 9.1. By
a computer-assisted enumeration, we verified that for every graph in
$\mathcal{C}_{0}$ with exactly six vertices of degree two, adding a copy of
$K_{1,3}$ and connecting them to form a $3$-regular graph of girth six results
either in an e-graph from $\mathcal{C}_{0}$ or in a non-critical e-graph.
Therefore, $F\neq G_{v}$.

If $G_{v}$ is disconnected and $F$ is a connected component of $G_{v}$, then
by Corollary 8.6, we have $|V(F)\cap N|\geq 3$ and $|(V(G_{v})\setminus
V(F))\cap N|\geq 3$. Since $|N|=6$, we have $|V(F)\cap N|=3$, and thus $F$ has
exactly three vertices of degree two. However, this contradicts (c0).
Therefore, $F$ is not a connected component of $G_{v}$, and thus by
Observation 2.6, $F$ contains a nailed vertex not belonging to $D$.

Suppose first that $F$ contains at least two nailed vertices; by (a0) and
Lemma 5.3, $F$ is a 5-cycle with exactly two nailed vertices. The three non-
nailed vertices of $F$ necessarily belong to $N\setminus D$. Since $G$ has
girth at least five, each of the vertices $v_{1}$, $v_{2}$, and $v_{3}$ has at
most one neighbor in $F$, and thus $|V(F)\cap N|\leq 3$. It follows that
$|V(F)\cap N|=3$. However, this implies that $(v,u)\in E(\vec{Z})$ for some
$u\in V(F)\cap N$, and thus $V(F)\cap D\neq\emptyset$ and
$|V(F)\cap(N\setminus D)|<3$, which is a contradiction.

Therefore, $F$ has exactly one nailed vertex $x$, and this vertex does not
belong to $D$; let $x^{\prime}$ be the neighbor of $x$ not in $F$. If both
$v_{1}$ and $v_{2}$ had two neighbors in $F$, then at most two edges of $G$
have exactly one end in either $V(F)\cup\\{v_{1},v_{2},v\\}$ (if $v_{3}$ has
no neighbor in $F$) or $V(F)\cup\\{v_{1},v_{2},v_{3},v\\}$ (if $v_{3}$ has a
neighbor in $F$), contradicting Corollary 8.6. By symmetry, we conclude that
at most one of the vertices $v_{1}$, $v_{2}$, and $v_{3}$ has two neighbors in
$F$, and in particular $|V(F)\cap N|\leq 4$.

$v$$v_{1}$$v_{2}$$v_{3}$$x$$x^{\prime}$$F$

$v$$v_{1}$$v_{2}$$v_{3}$$z$$x$$x^{\prime}$$F$

Figure 37: The e-graph $G_{1}$.

On the other hand, $F$ contains at least three non-nailed vertices of degree
two by (b0), and thus $|V(F)\cap N|\geq 3$. If each of the vertices $v_{1}$,
$v_{2}$, and $v_{3}$ has at least one neighbor in $F$, then let $S=(N\setminus
V(F))\cup\\{x^{\prime}\\}$ and let $G_{1}$ be the sub-e-graph of $G$
consisting of $G[V(F)\cup\\{v_{1},v_{2},v_{3},v\\}]$ and the edges from $S$ to
$V(F)\cup\\{v_{1},v_{2},v_{3},v\\}$. If one of them (say $v_{3}$) does not
have any neighbor in $F$, then let $z$ be the unique neighbor of $v_{1}$ or
$v_{2}$ distinct from $v$ and not belonging to $F$, let
$S=\\{v_{3},z,x^{\prime}\\}$ and let $G_{1}$ be the sub-e-graph of $G$
consisting of $G[V(F)\cup\\{v_{1},v_{2},v\\}]$ and the edges from $S$ to
$V(F)\cup\\{v_{1},v_{2},v\\}$, see Figure 37. The configuration $G_{1}$
attaching at $S$ is excluded by reducibility. ∎

Next, we constrain the difficult $5$-cycle graph, ensuring that not too many
vertices get nailed in the reductions according to the previous Lemma.

######  Lemma 10.2.

Let $G$ be a minimum counterexample and let $\vec{Z}$ be its difficult
$5$-cycle graph. Then $\vec{Z}$ has maximum indegree at most one.

###### Proof.

Suppose for a contradiction that $(v,u),(v^{\prime},u)\in E(\vec{Z})$ for two
distinct vertices $v,v^{\prime}\in V(G)$. Let $v_{1}$, $v_{2}$, and $v_{3}$ be
the neighbors of $v$, where $uv_{2}\in E(G)$. Let $K=uu_{1}u_{2}u_{3}u_{4}$ be
a $5$-cycle such that $u(v,K)=u$, and for $i\in\\{1,2,3\\}$, let $z_{i}$ be
the neighbor of $v_{i}$ distinct from $v$ and not belonging to $K$. Let
$K^{\prime}$ be a $5$-cycle such that $u(v^{\prime},K^{\prime})=u$. Let
$G_{0}$ be the sub-e-graph of $G$ consisting of $K$, $K^{\prime}$, and all
paths of length two from $v$ to $K$ and from $v^{\prime}$ to $K^{\prime}$. If
$G_{0}$ is an induced sub-e-graph, then let $S_{1}$ be the set of vertices in
$V(G)\setminus V(G_{0})$ with a neighbor in $G_{0}$ and let $G_{1}$ be the
sub-e-graph of $G$ consisting of $G_{0}$ and the edges between $S_{1}$ and
$G_{0}$.

$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{1}$$z_{3}$$z_{2}$$K$
Figure 38: The case $K[W(v,K)]$ is connected.
$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}=v^{\prime}$$z_{1}$$z_{3}$$z_{2}$
Figure 39: The e-graph $G_{1}$ in the case $K[W(v,K)]$ is connected and
$v^{\prime}=u_{3}$. Thick lines show the e-graph $G_{0}$.

Suppose first that $K[W(v,K)]$ is connected, and thus we can assume
$v_{1}u_{1},v_{3}u_{4}\in E(G)$, see Figure 38. The vertex $v^{\prime}\neq v$
is at distance two from $u$, and thus by symmetry we can assume
$v^{\prime}\in\\{v_{3},u_{3},z_{2}\\}$. Let us discuss the three cases
separately.

  * •

Suppose $v^{\prime}=v_{3}$. Then $u_{4}$ and $v$ are vertices with neighbors
in $K^{\prime}$ and $K^{\prime}$ contains the path $u_{1}uv_{2}$. Then
$u,v_{2}\in W(v^{\prime},K^{\prime})$ and $u(v^{\prime},K^{\prime})=u$, and
thus $u_{1}\in W(v^{\prime},K^{\prime})$. But $z_{3}u_{1}\not\in E(G)$ since
$G$ has girth at least five, which is a contradiction.

  * •

Suppose $v^{\prime}=u_{3}$. Then $u_{2}$ and $u_{4}$ have neighbors in
$K^{\prime}$, and thus $K^{\prime}=uv_{2}vv_{1}u_{1}$. Since $u,u_{1}\in
W(v^{\prime},K^{\prime})$ and $u(v^{\prime},K^{\prime})=u$, we have $v_{2}\in
W(v^{\prime},K^{\prime})$, and thus $u_{3}z_{2}\in E(G)$. Note that since $G$
has girth at least five, $G_{0}$ is an induced sub-e-graph of $G$. The e-graph
$G_{1}$ in this situation is depicted in Figure 39. But the configuration
$G_{1}$ attaching at $S_{1}$ is excluded by reducibility.

  * •

Suppose $v^{\prime}=z_{2}$. By Lemma 9.1, $z_{2}v_{2}u$ is the only path of
length two from $z_{2}$ to $u$, and thus $u_{1}uu_{4}\subset K^{\prime}$.
Using the fact that $G$ has girth at least five, observe that $K$ is the only
$5$-cycle containing this path; hence, we have $K=K^{\prime}$. Moreover,
$z_{2}v_{1},z_{2}v_{3}\not\in E(G)$, and thus
$W(v^{\prime},K)=\\{u,u_{2},u_{3}\\}$. For $i\in\\{2,3\\}$, let
$u^{\prime}_{i}$ be the common neighbor of $z_{2}$ and $u_{i}$.

$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{1}=u_{2}^{\prime\prime}$$z_{3}=u_{3}^{\prime\prime}$$z_{2}$$u_{3}^{\prime}$$u_{2}^{\prime}$$K=K^{\prime}$
Figure 40: The critical e-graph arising in the case $K[W(v,K)]$ is connected
and $v^{\prime}=z_{2}$.

Since $G$ has girth at least five, $z_{3}\neq u^{\prime}_{3}$ and $z_{1}\neq
u^{\prime}_{2}$. If $z_{3}=u^{\prime}_{2}$, then Corollary 8.6 implies
$z_{1}=u^{\prime}_{3}$; however, the resulting e-graph is $11/4$-colorable.
Hence $z_{3}\neq u^{\prime}_{2}$ and symmetrically $z_{1}\neq u^{\prime}_{3}$.
Consequently, $G_{0}$ is an induced sub-e-graph of $G$. For $i\in\\{2,3\\}$,
let $u^{\prime\prime}_{i}$ be the neighbor of $u^{\prime}_{i}$ distinct from
$u_{i}$ and $z_{2}$. If $u^{\prime\prime}_{2}=z_{1}$ and
$u^{\prime\prime}_{3}=z_{3}$, then $z_{1}z_{3}\in E(G)$ by Corollary 8.6 and
the resulting e-graph depicted in Figure 40 belongs to $\mathcal{C}_{0}$.
Hence, we can by symmetry assume $u^{\prime\prime}_{2}\neq z_{1}$.

$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{1}$$z_{3}$$z_{2}$$u_{3}^{\prime}$$u_{2}^{\prime}$$u_{2}^{\prime\prime}$$u_{3}^{\prime\prime}$$K=K^{\prime}$$z_{1}$$a$$b$$u_{2}^{\prime\prime}$$z_{3}$$u_{3}^{\prime\prime}$$G_{1}$$H_{2}$
Figure 41: The e-graph $G_{1}$ in the case $K[W(v,K)]$ is connected and
$v^{\prime}=z_{2}$, and the replacement e-graph $H_{2}$.

Let $H_{2}$ be the e-graph with the vertex set $S_{1}\cup\\{a,b\\}$, the edges
$z_{1}a$, $ab$, and $bu^{\prime\prime}_{2}$, and
$d_{H_{2}}(a)=d_{H_{2}}(b)=3$, see Figure 41. The standard argument for the
configuration $G_{1}$ attaching at $S_{1}$ and the replacement graph $H_{2}$
enforcing $|\varphi(z_{1})\cap\varphi(u^{\prime\prime}_{2})|\leq 3$ gives an
e-graph $G^{\prime}$ and its proper induced sub-e-graph
$F_{2}\in\mathcal{C}_{0}$; note that $F_{2}\neq G^{\prime}$ by (a0), since
$G^{\prime}$ has at least three nailed vertices.

Note that $G^{\prime}$ is connected by Corollary 8.6, and that $a,b\in
V(F_{2})$ since $F_{2}$ is not an induced sub-e-graph of the critical e-graph
$G$. By Observation 2.6, $F_{2}$ contains a nailed vertex distinct from the
nailed vertices $a$ and $b$, contradicting (a0).

$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{1}$$z_{3}$$z_{2}$$u_{1}^{\prime}$$u_{4}^{\prime}$$K$
Figure 42: The case $K[W(v,K)]$ is not connected.

Therefore, $K[W(v,K)]$ is not connected, and symmetrically,
$K^{\prime}[W(v^{\prime},K^{\prime})]$ is not connected. By symmetry, we can
assume $v_{1}u_{2},v_{3}u_{3}\in E(G)$. For $i\in\\{1,4\\}$, let
$u^{\prime}_{1}$ and $u^{\prime}_{4}$ be the neighbors of $u_{1}$ and $u_{4}$
distinct from $u$, $u_{2}$, and $u_{3}$, see Figure 42. The vertex
$v^{\prime}\neq v$ is at distance two from $u$, and thus by symmetry we can
assume $v^{\prime}\in\\{u_{3},z_{2},u^{\prime}_{4}\\}$. Let us discuss the
three cases separately.

  * •

Suppose $v^{\prime}=u_{3}$. Then $u,u_{1}\in W(v^{\prime},K^{\prime})$, and
since $u=u(v^{\prime},K^{\prime})$ and $u_{3}$ and $u_{1}$ have a common
neighbor, it follows that $K^{\prime}[W(v^{\prime},K^{\prime})]$ is connected,
which is a contradiction.

  * •

Suppose $v^{\prime}=z_{2}$. Then $u_{1}uu_{4}\subset K^{\prime}$. Since $G$
has girth at least five, observe that $z_{2}$ is not at distance two from
$u_{2}$ or $u_{3}$, and since $K^{\prime}[W(v^{\prime},K^{\prime})]$ is not
connected, we have $K^{\prime}\neq K$. Consequently,
$u^{\prime}_{1}u^{\prime}_{4}\in E(G)$,
$K^{\prime}=u^{\prime}_{1}u_{1}uu_{4}u^{\prime}_{4}$, and
$W(v^{\prime},K^{\prime})=\\{u^{\prime}_{1},u,u^{\prime}_{4}\\}$. Hence, for
$i\in\\{1,4\\}$, $z_{2}$ and $u^{\prime}_{i}$ has a common neighbor $w_{i}$.

$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$v^{\prime}=z_{2}$$u_{1}^{\prime}$$u_{4}^{\prime}$$w_{1}$$w_{4}$
Figure 43: The critical e-graph arising in the case $K[W(v,K)]$ is not
connected and $v^{\prime}=z_{2}$. It is isomorphic to $F_{14}^{(2)}$.
$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{1}$$z_{3}$$v^{\prime}=z_{2}$$u_{1}^{\prime}$$u_{4}^{\prime}$$w_{1}$$w_{4}$
Figure 44: The e-graph $G_{1}$ in the case $K[W(v,K)]$ is not connected and
$v^{\prime}=z_{2}$.

By Corollary 8.6, $\\{v_{1},v_{3},w_{1},w_{4}\\}$ is either an independent set
in $G$ or contains two edges with pairwise distinct endpoints. In the latter
case, one of the graphs arising in this way is $11/4$-colorable and the other
one (depicted in Figure 43) belongs to $\mathcal{C}_{0}$. Therefore, we can
assume $G_{0}$ is an induced sub-e-graph of $G$. The e-graph $G_{1}$ in this
case is depicted in Figure 44. However, the configuration $G_{1}$ attaching at
$S_{1}$ is excluded by reducibility.

  * •

Suppose $v^{\prime}=u^{\prime}_{4}$. Then $u_{1}uv_{2}\subset K^{\prime}$. The
vertices $v_{2}$ and $u_{2}$ do not have a common neighbor, and the vertices
$u_{1}$ and $v$ do not have a common neighbor, and thus
$z_{2}u^{\prime}_{1}\in E(G)$ and $K^{\prime}=z_{2}u^{\prime}_{1}u_{1}uv_{2}$.
Since $K^{\prime}[W(v^{\prime},K^{\prime})]$ is not connected,
$u^{\prime}_{4}$ has a common neighbor $y_{1}$ with $u^{\prime}_{1}$ and a
common neighbor $y_{2}$ with $z_{2}$.

$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{2}$$u_{1}^{\prime}$$v^{\prime}=u_{4}^{\prime}$$y_{2}$$y_{1}$
Figure 45: The critical e-graph arising in the case $K[W(v,K)]$ is not
connected and $v^{\prime}=u^{\prime}_{4}$. It is isomorphic to $F_{14}^{(1)}$.
$v$$v_{1}$$v_{2}$$v_{3}$$u$$u_{1}$$u_{2}$$u_{4}$$u_{3}$$z_{1}$$z_{3}$$z_{2}$$u_{1}^{\prime}$$v^{\prime}=u_{4}^{\prime}$$y_{2}$$y_{1}$
Figure 46: The e-graph $G_{1}$ in the case $K[W(v,K)]$ is not connected and
$v^{\prime}=u^{\prime}_{4}$.

By Corollary 8.6, $\\{v_{1},v_{3},y_{1},y_{2}\\}$ is either an independent set
in $G$ or contains two edges with pairwise distinct endpoints. In the latter
case, one of the graphs arising in this way is $11/4$-colorable and the other
one (depicted in Figure 45) belongs to $\mathcal{C}_{0}$. Therefore, we can
assume $G_{0}$ is an induced sub-e-graph of $G$. The e-graph $G_{1}$ in this
case is depicted in Figure 46. However, the configuration $G_{1}$ attaching at
$S_{1}$ is excluded by reducibility.

∎

We are now ready to prove our main result.

###### Proof of Lemma 2.4.

Suppose for a contradiction that there exists an e-graph $G\in\mathcal{C}$ not
belonging to $\mathcal{C}_{0}$; we can assume $G$ is a minimum counterexample.
Consequently, $G$ is $3$-regular by Lemma 8.5 and has girth at least five by
Lemma 9.1. Let $\vec{Z}$ be the difficult $5$-cycle graph of $G$; by Lemma
10.2, $\vec{Z}$ has maximum indegree at most one.

For each vertex $v\in V(G)$, let the e-graph $G_{v}$ be defined as in the
statement of Lemma 10.1 and let $\varphi_{v}$ be an $11/4$-coloring of
$G_{v}$. Let $v_{1}$, $v_{2}$, and $v_{3}$ be the neighbors of $v$ in $G$, and
let us extend $\varphi_{v}$ to a set $11$-coloring of $G$ as follows. For
$i\in\\{1,2,3\\}$, denoting by $u_{i,1}$ and $u_{i,2}$ the neighbors of
$v_{i}$ distinct from $v$, choose $\varphi_{v}(v_{i})$ as a subset of
$[0,11)\setminus(\varphi_{v}(u_{i,1})\cup\varphi_{v}(u_{i,2}))$ of measure
$1$. Then, select $\varphi_{v}(v)$ as a subset of
$[0,11)\setminus(\varphi_{v}(v_{1})\cup\varphi_{v}(v_{2})\cup\varphi_{v}(v_{3}))$
of measure $8$.

Let $n=|V(G)|$ and let $\varphi=\sum_{v\in V(G)}\tfrac{1}{n}\varphi_{v}$.
Consider a vertex $v\in V(G)$ whose indegree in $\vec{Z}$ is $d\leq 1$. We
have $|\varphi_{v}(v)|=8$ and for each neighbor $x$ of $v$, we have
$|\varphi_{x}(v)|=1$. For each vertex $u$ at distance two from $v$, we have
$|\varphi_{u}(v)|=4$ if $(u,v)\in E(\vec{Z})$ and $|\varphi_{u}(v)|=5$ if
$(u,v)\not\in E(\vec{Z})$. Finally, for each vertex $y$ at distance greater
than two from $v$, we have $|\varphi_{y}(v)|=4$. Since $G$ has girth five,
exactly $6$ vertices of $G$ are at distance exactly two from $v$. By
Observation 2.7, we have

$\displaystyle|\varphi(v)|$ $\displaystyle=\frac{(n-10)\times 4+8+3\times
1+(6-d)\times 5+d\times 4}{n}$ $\displaystyle=4+\frac{4-3\times
3+(6-d)}{n}\geq 4.$

Consequently, an $11/4$-coloring $\varphi^{\prime}$ of $G$ can be obtained
from $\varphi$ by choosing $\varphi^{\prime}(v)\subseteq\varphi(v)$ of measure
$4$ for every $v\in V(G)$. This is a contradiction. ∎

## References

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## Appendix

Here we list all elements of the set $\mathcal{C}$. The format is as follows.
The description of a graph is a sequence of parts of form “$a:xyz;$”, meaning
that the edges $ax$, $ay$, and $az$ should be added to the graph. The final
part of the sequence is of form “$x1y1$”, meaning that $d_{G}(x)=\deg x+1$ and
$d_{G}(y)=y+1$; all other vertices are non-nailed.

5 vertices 0 nails a:bd;b:c;c:e;d:e;

5 vertices 1 nails a:bd;b:c;c:e;d:e;d1

10 vertices 0 nails a:ei;b:cg;c:d;d:j;e:gh;f:ghi;h:j;

10 vertices 1 nails a:de;b:fj;c:dh;e:i;f:hi;g:hij;e1

15 vertices 0 nails a:en;b:go;c:di;d:m;e:ik;f:ikn;g:jl;h:jlo;j:m;k:l;

18 vertices 0 nails
a:bg;b:h;c:di;d:j;e:fk;f:l;g:im;h:in;j:oq;k:pr;l:qr;m:pq;n:op;o:r;

18 vertices 0 nails
a:fr;b:ch;c:k;d:ei;e:j;f:hl;g:hmr;i:np;j:op;k:no;l:nq;m:oq;p:q;

10 vertices 1 nails a:be;b:f;c:gj;d:ei;f:gh;g:i;h:ij;e1

15 vertices 0 nails a:en;b:go;c:im;d:kl;e:ik;f:ikn;g:jl;h:jlo;j:m;

10 vertices 1 nails a:bf;b:c;c:j;d:eh;e:i;f:hi;g:hij;e1

15 vertices 0 nails a:fn;b:ch;c:d;d:o;e:jk;f:jl;g:jln;h:km;i:kmo;l:m;

18 vertices 0 nails
a:bg;b:h;c:ei;d:fj;e:f;g:ko;h:mo;i:lp;j:np;k:nq;l:mr;m:n;o:r;p:q;q:r;

18 vertices 0 nails
a:fr;b:dh;c:ei;d:e;f:jn;g:knr;h:lo;i:mo;j:lp;k:mp;l:q;m:q;n:q;o:p;

12 vertices 0 nails a:gk;b:ci;c:e;d:fl;e:f;g:ij;h:ijk;j:l;

12 vertices 0 nails a:bi;b:l;c:dk;d:l;e:gj;f:hk;g:h;i:jk;j:l;

12 vertices 0 nails a:cg;b:ei;c:d;d:l;e:f;f:k;g:ij;h:ijl;j:k;

15 vertices 0 nails a:bh;b:i;c:dj;d:m;e:gk;f:gl;h:kn;i:ln;j:lo;k:o;m:no;

15 vertices 0 nails a:bg;b:i;c:jo;d:fk;e:fl;g:hm;h:no;i:km;j:kn;l:mn;

15 vertices 0 nails a:bg;b:i;c:jo;d:fk;e:fl;g:hm;h:no;i:km;j:ln;k:n;l:m;

15 vertices 0 nails a:ch;b:di;c:k;d:l;e:fj;f:n;g:mn;h:jo;i:mo;j:k;k:m;l:no;

15 vertices 0 nails a:bh;b:i;c:dj;d:k;e:fl;f:n;g:mn;h:jo;i:mo;j:l;k:lm;n:o;

15 vertices 0 nails a:dh;b:ci;c:j;d:l;e:fk;f:n;g:mn;h:io;i:k;j:km;l:no;m:o;

15 vertices 0 nails a:bh;b:i;c:ej;d:fk;e:f;g:lm;h:jn;i:ln;j:o;k:mo;l:o;m:n;

15 vertices 0 nails a:bh;b:i;c:ej;d:fk;e:f;g:lm;h:jn;i:ln;j:o;k:lo;m:no;

15 vertices 0 nails a:bh;b:i;c:ej;d:fk;e:f;g:lm;h:lo;i:mo;j:mn;k:no;l:n;

DANGEROUS! 5 vertices 2 nails a:bc;b:e;c:d;d:e;c1d1

DANGEROUS! 5 vertices 2 nails a:ce;b:cd;d:e;c1d1

7 vertices 1 nails a:bf;b:d;c:eg;d:e;f:g;f1

7 vertices 0 nails a:bf;b:d;c:eg;d:e;f:g;

9 vertices 0 nails a:bh;b:d;c:ei;d:f;e:g;f:g;h:i;

8 vertices 0 nails a:bh;b:ce;c:fg;d:egh;e:f;f:h;

8 vertices 0 nails a:ch;b:eg;c:ef;d:efh;f:g;

8 vertices 1 nails a:dh;b:cf;c:g;d:fg;e:fgh;c1

13 vertices 0 nails a:dl;b:fm;c:hk;d:hi;e:hil;f:jk;g:jkm;i:j;

13 vertices 1 nails a:dl;b:fm;c:hk;d:hi;e:hil;f:jk;g:jkm;i:j;c1

18 vertices 0 nails
a:dq;b:fr;c:ho;d:km;e:kmq;f:ln;g:lnr;h:jp;i:jop;j:k;l:p;m:n;

18 vertices 0 nails
a:er;b:cg;c:h;d:ij;e:im;f:jmr;g:kn;h:ln;i:o;j:p;k:oq;l:pq;m:n;o:p;

18 vertices 0 nails
a:be;b:g;c:hr;d:ij;e:im;f:jnr;g:km;h:ln;i:o;j:p;k:pq;l:oq;m:n;o:p;

18 vertices 0 nails
a:be;b:g;c:hr;d:ij;e:im;f:jnr;g:km;h:ln;i:o;j:p;k:oq;l:pq;m:n;o:p;

13 vertices 1 nails a:em;b:dg;c:hi;d:j;e:il;f:ilm;g:hk;h:j;j:k;k:l;d1

18 vertices 0 nails
a:dq;b:fr;c:hj;d:jn;e:jnq;f:ko;g:kor;h:il;i:mp;k:p;l:mo;m:n;

21 vertices 0 nails
a:eu;b:cg;c:h;d:ij;e:io;f:iou;g:kp;h:lp;j:mn;k:mr;l:ms;n:qs;o:q;p:t;q:r;r:t;s:t;

21 vertices 0 nails
a:eu;b:cg;c:h;d:ij;e:io;f:iou;g:kp;h:lp;j:mn;k:ms;l:st;m:q;n:rt;o:r;p:q;q:t;r:s;

21 vertices 0 nails
a:eu;b:cg;c:h;d:ij;e:io;f:iou;g:kp;h:lp;j:mn;k:mt;l:st;m:q;n:rt;o:r;p:q;q:s;r:s;

15 vertices 0 nails a:ho;b:dg;c:ej;d:m;e:m;f:kl;g:jn;h:ln;i:lno;j:k;k:m;

15 vertices 0 nails a:go;b:ci;c:l;d:ek;e:l;f:jk;g:jn;h:jno;i:km;l:m;m:n;

18 vertices 0 nails
a:bf;b:h;c:dg;d:i;e:jr;f:lq;g:mq;h:oq;i:mo;j:np;k:npr;l:mn;o:p;

18 vertices 0 nails
a:be;b:g;c:hq;d:ir;e:fk;f:lq;g:km;h:lm;i:no;j:nor;k:p;l:p;m:n;o:p;

18 vertices 0 nails
a:be;b:g;c:hq;d:ir;e:fm;f:nq;g:km;h:ln;i:op;j:opr;k:no;l:mp;

18 vertices 0 nails
a:gr;b:cf;c:i;d:km;e:jl;f:jq;g:mo;h:mor;i:nq;j:p;k:lp;l:q;n:op;

18 vertices 0 nails
a:fr;b:ch;c:i;d:jl;e:km;f:lo;g:lor;h:kq;i:nq;j:kp;m:pq;n:op;

18 vertices 0 nails
a:bf;b:g;c:hr;d:jl;e:km;f:jn;g:ln;h:mo;i:mor;j:p;k:lp;n:q;o:q;p:q;

18 vertices 0 nails
a:bf;b:g;c:hr;d:jk;e:lm;f:jq;g:kq;h:lo;i:lor;j:n;k:p;m:pq;n:op;

18 vertices 0 nails
a:bf;b:g;c:hr;d:kl;e:jm;f:jq;g:kq;h:lo;i:lor;j:n;k:p;m:pq;n:op;

18 vertices 0 nails
a:bf;b:g;c:hr;d:jl;e:km;f:jq;g:kq;h:lo;i:lor;j:p;k:p;m:nq;n:op;

DANGEROUS! 8 vertices 2 nails a:cg;b:eh;c:ef;d:efg;f:h;b1h1

DANGEROUS! 8 vertices 2 nails a:bd;b:f;c:gh;d:eg;e:fh;f:g;b1c1

16 vertices 1 nails
a:be;b:f;c:dg;d:h;e:jm;f:km;g:il;h:in;i:o;j:ln;k:no;l:p;m:p;o:p;d1

24 vertices 0 nails
a:de;b:cf;c:g;d:h;e:im;f:jn;g:kn;h:lm;i:ot;j:ps;k:qs;l:rt;m:u;n:v;o:su;p:vw;q:vx;r:ux;t:w;w:x;

24 vertices 0 nails
a:dx;b:cf;c:g;d:hl;e:ilx;f:jw;g:kw;h:mq;i:nq;j:or;k:pr;l:s;m:su;n:sv;o:tu;p:tv;q:r;t:w;u:v;

16 vertices 1 nails
a:be;b:f;c:dg;d:h;e:jm;f:km;g:il;h:in;i:o;j:ln;k:lo;m:p;n:p;o:p;d1

24 vertices 0 nails
a:bd;b:f;c:gx;d:hl;e:imx;f:jl;g:km;h:nr;i:os;j:pr;k:qs;l:t;m:u;n:ot;o:u;p:tw;q:uv;r:v;s:w;v:w;

24 vertices 0 nails
a:bd;b:f;c:gx;d:hl;e:imx;f:jl;g:km;h:nr;i:os;j:pr;k:qs;l:t;m:u;n:ot;o:u;p:tv;q:uv;r:w;s:w;v:w;

16 vertices 1 nails
a:be;b:f;c:dg;d:h;e:jm;f:km;g:il;h:in;i:o;j:lo;k:no;l:p;m:p;n:p;d1

10 vertices 1 nails a:cg;b:dh;c:j;d:j;e:fi;f:g;g:h;h:i;i:j;f1

18 vertices 0 nails
a:bh;b:i;c:eg;d:fj;e:k;f:k;g:lm;h:np;i:op;j:no;k:l;l:q;m:nr;o:q;p:r;q:r;

18 vertices 0 nails
a:bh;b:j;c:eg;d:fi;e:k;f:k;g:lm;h:nq;i:op;j:pq;k:l;l:r;m:np;n:r;o:qr;

18 vertices 0 nails
a:bh;b:j;c:eg;d:fi;e:k;f:k;g:lo;h:np;i:mq;j:pq;k:l;l:r;m:op;n:or;q:r;

FULL! 14 vertices 0 nails
a:dfn;b:ehn;c:gin;d:gj;e:fk;f:l;g:m;h:im;i:l;j:kl;k:m;

FULL! 14 vertices 0 nails
a:cgm;b:efm;c:ei;d:fjn;e:l;f:k;g:jk;h:iln;i:j;k:l;m:n;

10 vertices 1 nails a:bg;b:i;c:dh;d:i;e:fh;f:j;g:hj;i:j;f1

18 vertices 0 nails
a:bi;b:j;c:eg;d:fh;e:k;f:k;g:hl;h:m;i:np;j:op;k:q;l:or;m:no;n:q;p:r;q:r;

18 vertices 0 nails
a:hr;b:df;c:eg;d:j;e:j;f:gk;g:l;h:mo;i:nor;j:p;k:mq;l:nq;m:p;n:p;o:q;

16 vertices 0 nails
a:bd;b:f;c:gp;d:hl;e:imp;f:jl;g:km;h:kn;i:jo;j:k;l:o;m:n;n:o;

18 vertices 0 nails
a:fr;b:dh;c:ei;d:k;e:j;f:hl;g:imr;h:n;i:o;j:lo;k:mn;l:p;m:q;n:p;o:q;p:q;

18 vertices 0 nails
a:bg;b:i;c:ej;d:fh;e:k;f:l;g:hm;h:q;i:jp;j:o;k:no;l:pq;m:or;n:pr;q:r;

18 vertices 0 nails
a:cg;b:dh;c:i;d:j;e:fk;f:l;g:hp;h:q;i:mp;j:nr;k:or;l:qr;m:oq;n:op;

18 vertices 0 nails
a:bf;b:h;c:ir;d:ej;e:k;f:lp;g:mqr;h:np;i:oq;j:op;k:nq;l:mo;m:n;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:mq;i:nr;j:o;k:pr;l:ns;m:nt;o:ps;p:t;q:u;s:u;t:u;

21 vertices 0 nails
a:eu;b:cg;c:h;d:ij;e:io;f:jou;g:kp;h:lp;i:m;j:n;k:qr;l:qs;m:nr;n:s;o:q;p:t;r:t;s:t;

21 vertices 0 nails
a:bf;b:g;c:dh;d:i;e:jk;f:jp;g:kp;h:lq;i:mq;j:n;k:o;l:nu;m:tu;n:r;o:su;p:s;q:r;r:t;s:t;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:nq;i:mr;j:o;k:pr;l:ou;m:pt;n:tu;o:s;p:u;q:s;s:t;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:nq;i:mr;j:o;k:pr;l:pu;m:ot;n:tu;o:u;p:s;q:s;s:t;

21 vertices 0 nails
a:bf;b:g;c:dh;d:i;e:jk;f:jp;g:kp;h:lq;i:mq;j:n;k:o;l:rt;m:st;n:os;o:t;p:r;q:u;r:u;s:u;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:nq;i:mr;j:o;k:pr;l:mu;m:s;n:tu;o:pt;p:u;q:s;s:t;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:nq;i:mr;j:o;k:pr;l:mu;m:s;n:tu;o:pu;p:t;q:s;s:t;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:mq;i:nr;j:o;k:pr;l:ps;m:pt;n:ot;o:s;q:u;s:u;t:u;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:mq;i:nr;j:o;k:pr;l:os;m:ot;n:pt;p:s;q:u;s:u;t:u;

21 vertices 0 nails
a:bf;b:g;c:dh;d:i;e:jk;f:jp;g:kp;h:lq;i:mq;j:n;k:o;l:ns;m:nt;o:rt;p:r;q:u;r:s;s:u;t:u;

16 vertices 0 nails
a:dp;b:cf;c:g;d:hl;e:ilp;f:jo;g:ko;h:jn;i:kn;j:m;k:m;l:m;n:o;

18 vertices 0 nails
a:fr;b:dh;c:ei;d:j;e:k;f:hl;g:imr;h:p;i:q;j:np;k:oq;l:nq;m:op;n:o;

18 vertices 0 nails
a:cg;b:dh;c:k;d:l;e:fi;f:j;g:hn;h:o;i:mq;j:nq;k:pq;l:op;m:pr;n:r;o:r;

21 vertices 0 nails
a:bf;b:g;c:dh;d:i;e:jk;f:jp;g:kp;h:lq;i:mq;j:n;k:o;l:nt;m:tu;n:r;o:su;p:s;q:r;r:u;s:t;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:nq;i:mr;j:o;k:pr;l:ou;m:pu;n:tu;o:s;p:t;q:s;s:t;

21 vertices 0 nails
a:bf;b:i;c:dg;d:h;e:jk;f:jr;g:lq;h:nq;i:mr;j:o;k:pr;l:pu;m:ou;n:tu;o:t;p:s;q:s;s:t;

13 vertices 1 nails a:bf;b:h;c:dg;d:i;e:jk;f:jm;g:kl;h:km;i:lm;j:l;e1

18 vertices 0 nails
a:bh;b:i;c:dg;d:j;e:kn;f:lm;g:mq;h:kr;i:lr;j:qr;k:p;l:o;m:p;n:oq;o:p;

18 vertices 0 nails
a:bh;b:i;c:dg;d:j;e:kn;f:lm;g:kq;h:lr;i:mr;j:qr;k:p;l:o;m:p;n:oq;o:p;

18 vertices 0 nails
a:bh;b:i;c:dg;d:j;e:kn;f:lm;g:lq;h:kr;i:mr;j:qr;k:p;l:o;m:p;n:oq;o:p;

13 vertices 1 nails a:be;b:g;c:hm;d:ij;e:fk;f:lm;g:ik;h:il;j:kl;d1

18 vertices 0 nails
a:bf;b:h;c:ir;d:jk;e:lm;f:gp;g:qr;h:jp;i:kq;j:n;k:o;l:op;m:nq;n:o;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:gp;g:qr;h:jp;i:kq;j:n;k:o;l:op;m:nq;n:o;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jk;e:lm;f:gp;g:qr;h:jp;i:kq;j:n;k:o;l:np;m:oq;n:o;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jm;e:kl;f:gp;g:qr;h:jp;i:kq;j:n;k:o;l:np;m:oq;n:o;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:gp;g:qr;h:jp;i:kq;j:n;k:n;l:op;m:oq;n:o;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jm;e:kl;f:gp;g:qr;h:jp;i:kq;j:n;k:n;l:op;m:oq;n:o;

13 vertices 1 nails a:be;b:g;c:hm;d:ij;e:fk;f:lm;g:ik;h:jl;i:l;j:k;d1

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:im;g:km;h:jl;i:l;j:m;k:l;e1

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:im;g:km;h:il;j:lm;k:l;e1

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:il;g:jl;h:jm;i:m;k:lm;e1

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:im;g:jm;h:lm;i:k;j:l;k:l;e1

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:hm;g:im;h:l;i:k;j:lm;k:l;e1

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:hm;g:im;h:l;i:l;j:km;k:l;e1

10 vertices 0 nails a:bf;b:h;c:ij;d:eh;e:i;f:gi;g:hj;

10 vertices 1 nails a:bg;b:i;c:dh;d:j;e:fj;f:g;g:h;h:i;i:j;f1

15 vertices 0 nails a:bg;b:h;c:io;d:ek;e:m;f:lm;g:km;h:kn;i:ln;j:lno;

12 vertices 0 nails a:eh;b:il;c:fi;d:gj;e:k;f:k;g:k;h:ij;j:l;

12 vertices 0 nails a:bh;b:j;c:kl;d:fj;e:gk;f:g;h:ik;i:jl;

12 vertices 0 nails a:bh;b:j;c:jl;d:fh;e:gi;f:k;g:k;h:i;i:l;j:k;

15 vertices 0 nails a:bh;b:i;c:dj;d:m;e:fl;f:m;g:kl;h:ko;i:lo;j:no;k:n;m:n;

15 vertices 0 nails a:bi;b:j;c:dh;d:m;e:fl;f:m;g:kl;h:in;i:o;j:lo;k:no;m:n;

15 vertices 0 nails a:bh;b:j;c:di;d:m;e:fl;f:m;g:kl;h:io;i:n;j:ko;k:n;l:o;m:n;

15 vertices 0 nails a:bh;b:k;c:ei;d:fj;e:n;f:n;g:lm;h:mo;i:jl;j:m;k:no;l:o;

15 vertices 0 nails a:go;b:di;c:ej;d:m;e:m;f:kl;g:kn;h:lno;i:jk;j:l;m:n;

15 vertices 0 nails a:bi;b:j;c:eh;d:fk;e:n;f:n;g:lm;h:ko;i:lo;j:mo;k:m;l:n;

15 vertices 0 nails a:bh;b:i;c:ej;d:fk;e:n;f:n;g:lm;h:jo;i:lo;j:k;k:m;l:n;m:o;

15 vertices 0 nails a:bh;b:i;c:ej;d:fk;e:n;f:n;g:lm;h:jo;i:lo;j:k;k:l;m:no;

9 vertices 0 nails a:fi;b:eh;c:fh;d:gh;e:fg;g:i;

10 vertices 0 nails a:fj;b:df;c:eg;d:i;e:i;f:h;g:hj;h:i;

10 vertices 1 nails a:bg;b:i;c:dh;d:i;e:fh;f:g;g:j;h:j;i:j;f1

15 vertices 0 nails a:go;b:ci;c:l;d:ek;e:l;f:jk;g:jn;h:jno;i:mn;k:m;l:m;

12 vertices 0 nails a:bh;b:j;c:jl;d:fh;e:gi;f:g;h:k;i:kl;j:k;

15 vertices 0 nails a:bi;b:j;c:dh;d:m;e:fl;f:m;g:kl;h:kn;i:ko;j:lo;m:n;n:o;

15 vertices 0 nails a:bi;b:j;c:dh;d:m;e:fl;f:m;g:kl;h:kn;i:lo;j:no;k:o;m:n;

15 vertices 0 nails a:bh;b:j;c:di;d:m;e:fl;f:m;g:kl;h:kn;i:ko;j:no;l:n;m:o;

13 vertices 0 nails a:bf;b:h;c:dg;d:i;e:jk;f:jm;g:kl;h:km;i:lm;j:l;

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:im;g:lm;h:jl;i:k;j:m;k:l;e1

18 vertices 0 nails
a:gr;b:cf;c:i;d:km;e:jl;f:jq;g:mo;h:mor;i:pq;j:n;k:lp;l:q;n:op;

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:im;g:lm;h:il;j:km;k:l;e1

18 vertices 0 nails
a:fr;b:ch;c:i;d:jl;e:km;f:lo;g:lor;h:kq;i:pq;j:kp;m:nq;n:op;

15 vertices 0 nails a:bi;b:j;c:eh;d:fk;e:n;f:n;g:lm;h:mo;i:kl;j:km;l:o;n:o;

15 vertices 0 nails a:bh;b:k;c:ei;d:fj;e:f;g:lm;h:mn;i:lo;j:mo;k:no;l:n;

18 vertices 0 nails
a:cg;b:dh;c:i;d:j;e:lm;f:kn;g:lp;h:kr;i:mp;j:mr;k:o;l:q;n:qr;o:pq;

18 vertices 0 nails
a:cg;b:dh;c:i;d:j;e:km;f:ln;g:kp;h:lr;i:mp;j:mr;k:q;l:q;n:or;o:pq;

18 vertices 0 nails
a:bg;b:h;c:di;d:j;e:km;f:ln;g:lp;h:np;i:kr;j:mr;k:o;l:q;m:q;n:r;o:pq;

18 vertices 0 nails
a:bg;b:i;c:dh;d:j;e:km;f:ln;g:kr;h:lq;i:lr;j:oq;k:p;m:or;n:pq;o:p;

18 vertices 0 nails
a:bg;b:i;c:dh;d:j;e:km;f:ln;g:kr;h:lq;i:lr;j:oq;k:o;m:pr;n:pq;o:p;

18 vertices 0 nails
a:bg;b:h;c:di;d:j;e:mn;f:kl;g:kr;h:lr;i:mq;j:oq;k:o;l:p;m:r;n:pq;o:p;

18 vertices 0 nails
a:bh;b:i;c:dg;d:j;e:kn;f:lm;g:kq;h:lr;i:mr;j:oq;k:p;l:o;m:p;n:qr;o:p;

18 vertices 0 nails
a:bg;b:i;c:dh;d:j;e:ln;f:km;g:lp;h:kr;i:np;j:qr;k:o;l:q;m:nr;o:pq;

18 vertices 0 nails
a:bg;b:h;c:di;d:j;e:kl;f:mn;g:kp;h:lp;i:mr;j:qr;k:q;l:m;n:or;o:pq;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:jp;g:kqr;h:lp;i:mq;j:n;k:o;l:m;n:oq;o:p;

18 vertices 0 nails
a:bh;b:i;c:dg;d:j;e:kn;f:lm;g:lq;h:kr;i:mr;j:oq;k:p;l:p;m:q;n:or;o:p;

18 vertices 0 nails
a:bh;b:i;c:dg;d:j;e:kn;f:lm;g:lq;h:kr;i:mr;j:oq;k:o;l:p;m:q;n:pr;o:p;

18 vertices 0 nails
a:bg;b:h;c:di;d:j;e:kl;f:mn;g:kp;h:lp;i:mr;j:or;k:q;l:m;n:qr;o:pq;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:jp;g:kqr;h:np;i:nq;j:k;l:op;m:oq;n:o;

18 vertices 0 nails
a:cg;b:dh;c:i;d:j;e:kl;f:mn;g:kq;h:mr;i:oq;j:or;k:p;l:mq;n:pr;o:p;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:jp;g:kqr;h:np;i:nq;j:o;k:o;l:mp;m:q;n:o;

18 vertices 0 nails
a:bg;b:i;c:dh;d:j;e:kn;f:lm;g:kp;h:lr;i:np;j:or;k:q;l:q;m:nr;o:pq;

13 vertices 0 nails a:be;b:g;c:hm;d:ij;e:fk;f:lm;g:ik;h:il;j:kl;

15 vertices 0 nails a:bg;b:k;c:lo;d:ek;e:l;f:ij;g:im;h:ino;j:mn;k:m;l:n;

18 vertices 0 nails
a:cg;b:dh;c:j;d:i;e:kl;f:mn;g:kq;h:mr;i:or;j:pq;k:o;l:mq;n:pr;o:p;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:jp;g:kqr;h:np;i:oq;j:k;l:op;m:nq;n:o;

13 vertices 0 nails a:be;b:g;c:hm;d:ij;e:fk;f:lm;g:ik;h:jl;i:l;j:k;

15 vertices 0 nails a:bg;b:k;c:lo;d:ek;e:l;f:ij;g:im;h:jno;i:n;j:m;k:m;l:n;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:jp;g:kqr;h:np;i:oq;j:o;k:n;l:mp;m:q;n:o;

11 vertices 0 nails a:bj;b:ck;c:df;d:hj;e:ijk;f:gi;g:hk;h:i;

13 vertices 0 nails a:be;b:g;c:hm;d:il;e:ij;f:ikm;g:jl;h:kl;j:k;

13 vertices 1 nails a:be;b:g;c:hm;d:il;e:ij;f:ikm;g:jl;h:kl;j:k;d1

18 vertices 0 nails
a:be;b:g;c:hr;d:ip;e:ln;f:lor;g:mn;h:mo;i:kq;j:kpq;k:l;m:q;n:o;

13 vertices 1 nails a:bf;b:g;c:eh;d:ij;e:k;f:im;g:jm;h:jl;i:k;k:l;l:m;e1

18 vertices 0 nails
a:bf;b:g;c:hr;d:jl;e:km;f:jp;g:lp;h:mq;i:mqr;j:n;k:lo;n:oq;o:p;

15 vertices 0 nails a:bf;b:h;c:io;d:ej;e:n;f:jl;g:jmo;h:kl;i:km;k:n;l:m;

15 vertices 0 nails a:go;b:di;c:ej;d:e;f:kl;g:km;h:lmo;i:kn;j:ln;m:n;

15 vertices 0 nails a:fo;b:ch;c:i;d:jn;e:lm;f:jl;g:jmo;h:kl;i:km;k:n;

18 vertices 0 nails
a:bf;b:h;c:ir;d:jl;e:km;f:jn;g:kor;h:ln;i:mo;j:p;k:p;l:m;n:q;o:q;p:q;