Coloring count cones of planar graphs

TL;DR

This paper studies the mathematical structure of coloring extension counts in planar graphs, introduces a conjecture related to the Four Color Theorem, and provides supporting evidence for it.

Contribution

It characterizes the coloring count cone for certain planar graphs, formulates a stronger conjecture than the Four Color Theorem, and offers evidence supporting this conjecture.

Findings

The coloring count function belongs to a specific cone in the function space.

A new conjecture strengthening the Four Color Theorem is proposed.

Evidence supporting the conjecture is presented.

Abstract

For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^\star_G$ denote the function that to each $4$-coloring $\psi$ of $C$ assigns the number of ways $\psi$ extends to a $4$-coloring of $G$. The block-count reducibility argument (which has been developed in connection with attempted proofs of the Four Color Theorem) is equivalent to the statement that the function $n^\star_G$ belongs to a certain cone in the space of all functions from $4$-colorings of $C$ to real numbers. We investigate the properties of this cone for $|C|=5$, formulate a conjecture strengthening the Four Color Theorem, and present evidence supporting this conjecture.

Equations

$$\sum _{\psi }{n}_{{\tilde{G}}_1}(\psi )n_{\widetilde{G_2}}(\psi ),$$

$$\{(\mu (C),\sigma (C)):\text{C is an ij-Kempe chain of φ}\}.$$

$$n_{\tilde{G},S}({\psi }_1)=n_{\tilde{G},S}({\psi }_2).$$

$$n_{\tilde{G}}(\psi )=\sum _{S\in {\mathcal{S}}_{\psi ,ij}}{n}_{\tilde{G},S}(\psi )=\sum _{S\in {\mathcal{S}}_{\psi ,ij}}{n}_{\tilde{G},S}.$$

$$n_{\tilde{G}}\in B_{d(\tilde{G})}.$$

$$\sum _{\psi }{n}_{\tilde{G}}{n}_{{\tilde{C}}_5}(\psi )=c\sum _{\psi }{n}_{{\tilde{R}}_{5,12}}{n}_{{\tilde{C}}_5}(\psi )=0.$$

$$n_{\tilde{G}}=\sum _i{c}_i{n}_{{\tilde{R}}_{d,i}}.$$

$$\sum _{\psi }{n}_{\tilde{G}}(\psi )n_{{\tilde{C}}_d}(\psi )\ge c_k\sum _{\psi }{n}_{{\tilde{R}}_{d,k}}(\psi )n_{{\tilde{C}}_d}(\psi )\ge c_k{n}_{{\tilde{R}}_{d,k}}({\psi }_0)n_{{\tilde{C}}_d}({\psi }_0)>0.$$

$$3\sum _{i=0}^4{n}_{\tilde{G}}({\psi }_i^{5,a})\ge \sum _{i=0}^4{n}_{\tilde{G}}({\psi }_i^{5,b}).$$

$$y(\psi )=\sum _{{\psi }_1,{\psi }_2:{\gamma }_k({\psi }_1,{\psi }_2)=\psi }{x}_1({\psi }_1)x_2({\psi }_2),$$

$$n_{{\tilde{G}}_c}=\sum _i{c}_i{n}_{{\tilde{R}}_{k,i}},$$

$$X(G)$$

$$|E(G)|=7+(7+b_1)+b_1+(b_1+2b_2+b_3)+b_2+b_3+b_4=14+3b_1+3b_2+2b_3+b_4.$$

Full text

Key words: graph coloring, planar graphs, four color theorem, coloring count cone

By the Four Color Theorem [1, 2, 5], every planar graph is $4$-colorable. Nevertheless, many natural followup questions regarding $4$-colorability of planar graphs are wide open. Even very basic precoloring extension questions, such as the one given in the following problem, are unresolved (a near-triangulation is a connected plane graph in which all faces except for the outer one have length three).

Note that there exist infinitely many near-triangulations $G$ with the outer face bounded by a $4$-cycle $C$ such that not every precoloring of $C$ extends to a $4$-coloring of $G$; and we do not have any good guess at how the near-triangulations with this property could be described.

Nevertheless, we do have some information about the precoloring extension properties of plane near-triangulations. For a plane near-triangulation $G$ with the outer face bounded by a cycle $C$, let $n^{\star}_{G}$ denote the function that to each $4$-coloring $\psi$ of $C$ assigns the number of ways $\psi$ extends to a $4$-coloring of $G$; hence, $\psi$ extends to a $4$-coloring of $G$ if and only if $n^{\star}_{G}(\psi)\neq 0$. Suppose $C=v_{1}v_{2}v_{3}v_{4}$ is a $4$-cycle and $\psi_{1}$, $\psi_{2}$ and $\psi_{3}$ are its $4$-colorings such that $\psi_{i}(v_{j})=j$ for $i\in\{1,2,3\}$ and $j\in\{1,2\}$, $\psi_{1}(v_{3})=\psi_{3}(v_{3})=1$, $\psi_{2}(v_{3})=3$, $\psi_{1}(v_{4})=\psi_{2}(v_{4})=2$, and $\psi_{3}(v_{4})=4$; see Figure 1. A standard Kempe chain argument shows that if $n^{\star}_{G}(\psi_{1})\neq 0$, then $n^{\star}_{G}(\psi_{2})\neq 0$ or $n^{\star}_{G}(\psi_{3})\neq 0$.

Actually, much more information can be obtained along these lines, using the idea of Block-count reducibility [3, 4] developed in connection with the attempts to prove the Four Color Theorem: Certain inequalities between linear combinations of $n^{\star}_{G}(\psi_{1})$, $n^{\star}_{G}(\psi_{2})$, and $n^{\star}_{G}(\psi_{3})$ are satisfied for all near-triangulations $G$, or equivalently, the vector $(n^{\star}_{G}(\psi_{1}),n^{\star}_{G}(\psi_{2}),n^{\star}_{G}(\psi_{3}))$ is contained in a certain cone in $\mathbb{R}^{3}$. The main goal of this note is to present and motivate a conjecture regarding this cone in the case of near-triangulations with the outer face bounded by a $5$-cycle; this conjecture strengthens the Four Color Theorem. We also provide evidence supporting this conjecture.

1 Definitions

In order to describe the cone we alluded to in the introduction, we need a number of definitions, which we introduce in this section. It is easier to state the idea in the dual setting of $3$-edge-colorings of cubic plane graphs, which is well-known to be equivalent to $4$-coloring of plane triangulations [6].

Some graphs in this paper may have parallel edges or loops. We call two parallel edges a double edge and three parallel edges a triple edge.

1.1 Near-cubic graphs and their edge-colorings

Let $G$ be a connected graph and let $v$ be a vertex of $G$. A half-edge is $(e,u)$, where $e$ is an edge and $u$ is one of its endpoints. If $e=uv$, when we say $u$ is incident with $(e,u)$ but it is not incident with $(e,v)$. We consider each edge $e=uv$ of $G$ as consisting of two half-edges $(e,u)$ and $(e,v)$ even if $e$ is a loop. Let $\nu$ be a bijection between the half-edges incident with $v$ and $\{0,\ldots,\deg(v)-1\}$ (so, if $v$ is incident with a loop, each half of the loop is assigned a different number by $\nu$). If all vertices of $G$ other than $v$ have degree three, we say that $\tilde{G}=(G,v,\nu)$ is a near-cubic graph. We say that $\tilde{G}$ is a plane near-cubic graph if $G$ is a plane graph and the half-edges incident with $v$ are drawn around it in the clockwise cyclic order $\nu^{-1}(0)$, …, $\nu^{-1}(\deg(v)-1)$. We define $d(\tilde{G})=\deg(v)$.

A $3$-edge-coloring of $\tilde{G}$ is an assignment of colors $1$, $2$, and $3$ to edges of $G$ such that any two edges incident with a common vertex other than $v$ have different colors. For an integer $d\geq 2$, a function $\psi:\{0,\ldots,d-1\}\to\{1,2,3\}$ is a $d$-precoloring if $|\psi^{-1}(1)|\equiv|\psi^{-1}(2)|\equiv|\psi^{-1}(3)|\equiv d\pmod{2}$. This parity condition is necessary, see Observation 2. We say that a $3$-edge-coloring $\varphi$ of $\tilde{G}$ extends a $d(\tilde{G})$-precoloring $\psi$ if for any edge $e$ incident with $v$ and a half-edge $h$ of $e$ incident with $v$, we have $\varphi(e)=\psi(\nu(h))$.

Observation 2**.**

For an integer $d\geq 2$, if a function $\psi:\{0,\ldots,d-1\}\to\{1,2,3\}$ does not satisfy $|\psi^{-1}(1)|\equiv|\psi^{-1}(2)|\equiv|\psi^{-1}(3)|\equiv d\pmod{2}$, then there is no $\tilde{G}$ and 3-edge-coloring $\varphi$ of $\tilde{G}$ such that $\varphi$ extends $\psi$.

Proof.

Let $\varphi$ be a 3-edge-coloring of $\tilde{G}$ extending $\psi$. Let $n=|V(\tilde{G})|$. Since $\tilde{G}$ is cubic except for one vertex of degree $d$, the handshaking lemma gives $|E(\tilde{G})|=(3(n-1)+d)/2$. For $i\in\{1,2,3\}$, then number of edges colored by $|\varphi^{-1}(i)|=((n-1)+|\psi^{-1}(i)|)/2$. Therefore the parities of $d$, $|\psi^{-1}(1)|$, $|\psi^{-1}(2)|$, and $|\psi^{-1}(3)|$ are the same. ∎

Let $n_{\tilde{G}}(\psi)$ denote the number of $3$-edge-colorings of $\tilde{G}$ which extend $\psi$. Via the theory of nowhere-zero flows [7], it is easy to establish the following correspondence between $4$-colorings of near-triangulations and $3$-edge-colorings in their duals. Recall $n^{\star}_{G}(\psi)$ denotes the number of $4$-colorings of $G$ which extend $\psi$.

Observation 3**.**

Let $\tilde{G}=(G,v,\nu)$ be a plane near-cubic graph, and let $G^{\star}$ be the dual of $G$ drawn so that the outer face of $G^{\star}$ corresponds to $v$. Suppose the outer face of $G^{\star}$ is bounded by a cycle $C$. Then there exists a mapping $f$ from $4$-colorings of $C$ to $d(\tilde{G})$-precolorings such that

• •

$f$* maps exactly four distinct $4$-colorings of $C$ to each $d(\tilde{G})$-precoloring, and*

• •

every $4$-coloring $\psi$ of $C$ satisfies $n^{\star}_{G^{\star}}(\psi)=n_{\tilde{G}}(f(\psi))$.

Given two near-cubic graphs $\tilde{G}_{1}=(G_{1},v_{1},\nu_{1})$ and $\tilde{G}_{2}=(G_{2},v_{2},\nu_{2})$ with $\deg(v_{1})=\deg(v_{2})$, let $\tilde{G}_{1}\oplus\tilde{G}_{2}$ denote the graph obtained from $G_{1}$ and $G_{2}$ by, for $0\leq i\leq\deg(v_{1})-1$, removing the half-edges $\nu_{1}^{-1}(i)$ and $\nu_{2}^{-1}(i)$ and connecting the other halfs of the edges. Note that $\tilde{G}_{1}\oplus\tilde{G}_{2}$ is a cubic graph, and if $\tilde{G}_{1}$ and $\tilde{G}_{2}$ are plane near-cubic graphs, then $\tilde{G}_{1}\oplus\tilde{G}_{2}$ is a cubic planar graph. Observe that the number of $3$-edge-colorings of $\tilde{G}_{1}\oplus\tilde{G}_{2}$ is

[TABLE]

where the sum goes over all $\deg(v_{1})$-precolorings $\psi$. For any integer $n\geq 3$, let $\tilde{C}_{n}$ denote the plane near-cubic graph $(W_{n},v,\nu)$, where $W_{n}$ is the wheel with the central vertex $v$ adjacent to all vertices of an $n$-cycle; see Figure 2.

1.2 Signatures and Kempe chains

The following definition of $d$-signature is will be used to capture the possible parities of 2-edge-colored cycles containing $v$ in a 3-edge-coloring of $\tilde{G}=(G,v,\nu)$ distinguished by the half-edges contained in the cycles. In particular, parity will be $s\in\{-1,1\}$ and the pair of half-edges will be $m$. For an integer $d\geq 2$, a $d$-signature is a set $S$ of pairs $(m,s)$, where $m$ is an unordered pair of integers in $\{0,\ldots,d-1\}$ and $s\in\{-1,1\}$, satisfying the following conditions:

• (i)

for any distinct $(m_{1},s_{1}),(m_{2},s_{2})\in S$ we have $m_{1}\cap m_{2}=\emptyset$, and

• (ii)

$S$ does not contain elements $(\{a,b\},s_{1})$ and $(\{c,d\},s_{2})$ such that $a<c<b<d$.

A $d$-precoloring $\psi$ is compatible in (distinct) colors $i,j\in\{1,2,3\}$ with a $d$-signature $S$ if

• •

$\psi^{-1}(\{i,j\})=\bigcup_{(m,s)\in S}m$, and

• •

for each $(\{a_{1},a_{2}\},s)\in S$, $\psi(a_{1})=\psi(a_{2})$ holds if and only if $s=-1$.

Now, consider a $3$-edge-coloring $\varphi$ of a near-cubic graph $\tilde{G}=(G,v,\nu)$. Each vertex other than $v$ is incident with edges of all three colors. Hence, for any distinct $i,j\in\{1,2,3\}$, the subgraph $G_{ij}$ of $G$ consisting of edges of colors $i$ or $j$ is a union of pairwise edge-disjoint cycles, vertex-disjoint except for possible intersections in $v$. An $ij$-Kempe chain of $\varphi$ is a cycle $C$ in $G_{ij}$ containing $v$; the sign $\sigma(C)$ of the $ij$-Kempe chain $C$ is $1$ if the length of $C$ is even and $-1$ if the length of $C$ is odd. If $h_{1}$ and $h_{2}$ are the half-edges in $C$ incident with $v$, we let $\mu(C)=\{\nu(h_{1}),\nu(h_{2})\}$. The $ij$-Kempe chain signature $\sigma_{ij}(\varphi)$ of $\varphi$ is defined as

[TABLE]

Note that if $\tilde{G}$ is plane, then the $ij$-Kempe chains do not cross and the $ij$-Kempe chain signature of $\varphi$ satisfies the condition (ii); and thus $\sigma_{ij}(\varphi)$ is a $d(\tilde{G})$-signature.

2 Coloring count cones

Let $\tilde{G}=(G,v,\nu)$ be a plane near-cubic graph and let $\psi$ be a $d(\tilde{G})$-precoloring. Suppose that $\psi$ is compatible (in colors $i,j\in\{1,2,3\}$) with a $d(\tilde{G})$-signature $S$. We define $n_{\tilde{G},S}(\psi)$ as the number of $3$-edge-colorings $\varphi$ of $\tilde{G}$ extending $\psi$ such that $\sigma_{ij}(\varphi)=S$. Note that swapping the colors $i$ and $j$ on any set of $ij$-Kempe chains of $\varphi$ results in another $3$-edge-coloring with the same $ij$-Kempe chain signature. Furthermore, clearly for any permutation $\pi$ of colors, we have $n_{\tilde{G},S}(\psi\circ\pi)=n_{\tilde{G},S}(\psi)$. This establishes bijections implying the following.

Observation 4**.**

Let $\tilde{G}$ be a plane near-cubic graph and let $S$ be a $d(\tilde{G})$-signature. Any $d(\tilde{G})$-precolorings $\psi_{1}$ and $\psi_{2}$ compatible with $S$ satisfy

[TABLE]

Hence, we can define an integer $n_{\tilde{G},S}$ to be equal to $n_{\tilde{G},S}(\psi)$ for an arbitrarily chosen $d(\tilde{G})$-precoloring $\psi$ compatible with $S$.

Let $d\geq 2$ be an integer and let $i,j\in\{1,2,3\}$ be distinct colors. For a $d$-precoloring $\psi$, let us define $\mathcal{S}_{\psi,ij}$ as the set of $d$-signatures compatible with $\psi$ in colors $ij$. Since every $3$-edge-coloring of $\tilde{G}$ has an $ij$-Kempe chain signature, we have

[TABLE]

Let $\mathcal{P}_{d}$ denote the set of all $d$-precolorings and $\mathcal{S}_{d}$ the set of all $d$-signatures. We will work in the vector spaces $\mathbb{R}^{\mathcal{P}_{d}}$ and $\mathbb{R}^{\mathcal{S}_{d}}$ with coordinates corresponding to the $d$-precolorings and to the $d$-signatures, respectively. For each integer $d\geq 2$, the coloring count cone $B_{d}$ is the set of all $x\in\mathbb{R}^{\mathcal{P}_{d}}$ such that

• •

$x(\psi)\geq 0$ for every $d$-precoloring $\psi$, and

• •

there exists $y\in\mathbb{R}^{\mathcal{S}_{d}}$ such that

• –

$y(S)\geq 0$ for every $d$-signature $S$, and

• –

$x(\psi)=\sum_{S\in\mathcal{S}_{\psi,ij}}y(S)$ for every $d$-precoloring $\psi$ and distinct colors $i,j\in\{1,2,3\}$.

Note that $B_{d}$ is indeed a cone, i.e., an unbounded polytope closed under linear combinations with non-negative coefficients. By (2), the vector of precoloring extension counts for any plane near-cubic graph belongs to the corresponding coloring count cone.

Theorem 5**.**

For each plane near-cubic graph $\tilde{G}$, we have

[TABLE]

Each cone is uniquely determined as the set of non-negative linear combinations of its rays. For $d\in\{2,3,4,5\}$, the rays of $B_{d}$ are easy to enumerate by hand or using polytope-manipulation software such as Sage Math or the Parma Polyhedra Library (a program doing so for $d=5$ can be found at http://lidicky.name/pub/4cone/). For a near-cubic graph $\tilde{G}$ such that $n_{\tilde{G}}$ is not the zero function, let $\mathrm{ray}(\tilde{G})$ denote the set of all non-negative multiples of $n_{\tilde{G}}$. Graphs $\tilde{R}_{2,1},\ldots,\tilde{R}_{5,12}$ used in the following lemma are depicted in Figure 3.

Lemma 6**.**

Refering to graphs in Figure 3:

• •

the cone $B_{2}$ has exactly one ray equal to $\mathrm{ray}(\tilde{R}_{2,1})$;

• •

the cone $B_{3}$ has exactly one ray equal to $\mathrm{ray}(\tilde{R}_{3,1})$;

• •

the cone $B_{4}$ has exactly four rays equal to $\mathrm{ray}(\tilde{R}_{4,1})$, …, $\mathrm{ray}(\tilde{R}_{4,4})$; and

• •

the cone $B_{5}$ has exactly $12$ rays equal to $\mathrm{ray}(\tilde{R}_{5,1})$, …, $\mathrm{ray}(\tilde{R}_{5,12})$.

Let us remark that $B_{6}$ has 208 rays; the direct method we employ is too slow to enumerate all rays for $d\geq 7$ on current workstations.

3 The cone $B_{5}$ and the conjecture

Note that while $\tilde{R}_{5,1}$, …, $\tilde{R}_{5,11}$ are plane, $\tilde{R}_{5,12}$ is not. Indeed, the following holds.

Lemma 7**.**

The following claims are equivalent.

• (a)

Every planar cubic $2$-edge-connected graph is $3$-edge-colorable.

• (b)

For every plane near-cubic graph $\tilde{G}$ with $d(\tilde{G})=5$, if $n_{\tilde{G}}\in\mathrm{ray}(\tilde{R}_{5,12})$, then $n_{\tilde{G}}$ is the zero function.

Proof.

Let us first prove that (a) implies (b). Consider a plane near-cubic graph $\tilde{G}=(G,v,\nu)$ such that $n_{\tilde{G}}\in\mathrm{ray}(\tilde{R}_{5,12})$, and thus for some constant $c\geq 0$, we have $n_{\tilde{G}}(\psi)=c\cdot n_{\tilde{R}_{5,12}}(\psi)$ for every $5$-precoloring $\psi$. Observe that $n_{\tilde{R}_{5,12}}(\psi)n_{\tilde{C}_{5}}(\psi)=0$ for every $5$-precoloring $\psi$ (since $\tilde{R}_{5,12}\oplus\tilde{C}_{5}$ is the Petersen graph, which is not $3$-edge-colorable; see Figure 4), and thus the number of $3$-edge-colorings of $\tilde{G}\oplus\tilde{C}_{5}$ using (1) is

[TABLE]

Hence, the planar cubic graph $\tilde{G}\oplus\tilde{C}_{5}$ is not $3$-edge-colorable. By (a), $\tilde{G}\oplus\tilde{C}_{5}$ has a bridge, and thus $G$ has a bridge. But then a standard parity argument implies that $\tilde{G}$ has no $3$-edge-coloring, and thus $n_{\tilde{G}}$ is the zero function.

Next, let us prove that (b) implies (a). Suppose for a contradiction that (b) holds, but there exists a plane cubic $2$-edge-connected graph that is not $3$-edge-colorable, and let $H$ be one with the smallest number of vertices. By Euler’s formula and possible parallel edges, $H$ has a face $f$ of length $2\leq d\leq 5$; hence, we can write $H=\tilde{G}\oplus\tilde{C}_{d}$ for a plane near-cubic graph $\tilde{G}$. By Theorem 5, we have $n_{\tilde{G}}\in B_{d}$, and by Lemma 6, there exist non-negative real numbers $c_{i}$ such that

[TABLE]

Observe there exists a plane near-cubic graph $\tilde{P}$ with $d-1$ vertices such that $\tilde{G}\oplus\tilde{P}$ is $2$-edge-connected. By the minimality of $H$, $\tilde{G}\oplus\tilde{P}$ is $3$-edge-colorable, and in particular $n_{\tilde{G}}$ is not the zero function. By (b), $n_{\tilde{G}}$ is not a positive multiple of $n_{\tilde{R}_{5,12}}$, and thus there exists an index $k\leq 11$ such that $c_{k}>0$. Observe that $\tilde{R}_{d,k}\oplus\tilde{C}_{d}$ is $3$-edge-colorable, and thus there exists a $d$-precoloring $\psi_{0}$ such that $n_{\tilde{R}_{d,k}}(\psi_{0})n_{\tilde{C}_{d}}(\psi_{0})>0$. However, then the number of $3$-edge-colorings of $H$ is

[TABLE]

This contradicts the assumption that $H$ is not $3$-edge-colorable. ∎

Note that (a) from Lemma 7 is well-known to be equivalent to the Four Color Theorem [6], and thus indeed there is no plane near-cubic graph $\tilde{G}$ with $d(\tilde{G})=5$ such that $n_{\tilde{G}}$ is not the zero function and $n_{\tilde{G}}\in\mathrm{ray}(\tilde{R}_{5,12})$; and furthermore, a direct proof of this fact would imply the Four Color Theorem. Motivated by this observation (and experimental evidence), we propose the following conjecture, a strengthening of the Four Color Theorem. Let $B^{\prime}_{5}$ denote the cone in $\mathbb{R}^{\mathcal{P}_{d}}$ with rays $\mathrm{ray}(\tilde{R}_{5,1})$, …, $\mathrm{ray}(\tilde{R}_{5,11})$.

Conjecture 8**.**

Every plane near-cubic graph $\tilde{G}$ with $d(\tilde{G})=5$ satisfies $n_{\tilde{G}}\in B^{\prime}_{5}$.

For $i\in\{0,\ldots,4\}$, let $\psi^{5,a}_{i}$ and $\psi^{5,b}_{i}$ denote the $5$-precolorings whose values at $j\in\{0,\ldots,4\}$ are defined by the following table; see also Figure 5. Notice that $i$ is a rotating the coloring.

[TABLE]

Note that each $5$-precoloring is obtained from one of these ten by a permutation of colors. The cone $B^{\prime}_{5}$ has exactly one facet which is not also a facet of $B_{5}$, giving an equivalent formulation of Conjecture 8.

Conjecture 9**.**

Every plane near-cubic graph $\tilde{G}$ with $d(\tilde{G})=5$ satisfies

[TABLE]

In the rest of the note, we provide some evidence supporting Conjecture 8; in particular, we show there are no counterexamples to the conjecture for plane near-cubic graphs with less than 30 vertices.

4 Evidence

In this section we present experimental evidence for the validity of Conjecture 8. Our goal is to show Corollary 20 stating that Conjecture 8 holds for graphs on at most 30 vertices. The main idea of our approach is to generate larger graphs $\tilde{G}$ from smaller graphs by planarity preserving operations. One such is depicted in Figure 7. We will generate “all” possibilities for $d(\tilde{G})\leq 7$ and particular ones with $d(\tilde{G})=8$. We then argue that all graphs in at most 30 vertices can be generated this way.

We begin by stating a few more definitions. A vector $x\in\mathcal{P}_{d}$ is invariant with respect to permutation of colors if all $d$-precolorings $\psi$ and $\psi^{\prime}$ that only differ by a permutation of colors satisfy $x(\psi)=x(\psi^{\prime})$.

See Figure 6 for an illustration of the following definitions. The rotation by $t$ of a $d$-precoloring $\psi$ is the $d$-precoloring $r_{t}(\psi)$ such that $r_{t}(\psi)((i+t)\bmod d)=\psi(i)$ for $i\in\{0,\ldots,d-1\}$. The flip of a $d$-precoloring $\psi$ is the $d$-precoloring $f(\psi)$ such that $f(\psi)(i)=\psi(d-1-i)$ for $i\in\{0,\ldots,d-1\}$. For $x\in\mathbb{R}^{\mathcal{P}_{d}}$, let $r_{t}(x)$ be defined as $y\in\mathbb{R}^{\mathcal{P}_{d}}$ such that $y(r_{t}(\psi))=x(\psi)$ for every $d$-precoloring $\psi$, and let $f(x)$ be defined as $z\in\mathbb{R}^{\mathcal{P}_{d}}$ such that $z(f(\psi))=x(\psi)$ for every $d$-precoloring $\psi$. A set $K\subseteq\mathbb{R}^{\mathcal{P}_{d}}$ is closed under rotations and flips if we have $x\in K$ if and only if $f(x)\in K$ and $r_{t}(x)\in K$ for all $t\in\{0,1,\ldots,d-1\}$. For a near-cubic graph $\tilde{G}=(G,v,\nu)$ with $\deg(v)=d$, let $r_{t}(\tilde{G})$ denote the near-cubic graph $(G,v,\nu_{1})$, where $\nu_{1}^{-1}((i+t)\bmod d)=\nu^{-1}(i)$ for $i\in\{0,\ldots,d-1\}$, and let $f(\tilde{G})$ denote the near-cubic graph $(G,v,\nu_{2})$, where $\nu_{2}^{-1}(i)=\nu_{2}^{-1}(d-1-i)$ for $i\in\{0,\ldots,d-1\}$.

Observation 10**.**

Let $\tilde{G}$ be a near-cubic graph, $d=d(\tilde{G})$ and $t\in\{0,\ldots,d-1\}$. Then $n_{r_{t}(\tilde{G})}=r_{t}(n_{\tilde{G}})$ and $n_{f(\tilde{G})}=f(n_{\tilde{G}})$.

Let $\psi_{1}$ be a $d_{1}$-precoloring and $\psi_{2}$ a $d_{2}$-precoloring. For an integer $k\leq\min(d_{1},d_{2})$, we say that $\psi_{1}$ $k$-matches $\psi_{2}$ if $\psi_{1}(d_{1}-k+i)=\psi_{2}(d_{2}-1-i)$ for $i\in\{0,1,\ldots,k-1\}$. By $\gamma_{k}(\psi_{1},\psi_{2})$, we denote the $(d_{1}+d_{2}-2k)$-precoloring $\gamma$ such that $\gamma(i)=\psi_{1}(i)$ for $i\in\{0,\ldots,d_{1}-k-1\}$ and $\gamma(i)=\psi_{2}(i-(d_{1}-k))$ for $i\in\{d_{1}-k,\ldots,d_{1}+d_{2}-2k-1\}$. For $x_{1}\in\mathbb{R}^{\mathcal{P}_{d_{1}}}$ and $x_{2}\in\mathbb{R}^{\mathcal{P}_{d_{2}}}$, we define $\gamma_{k}(x_{1},x_{2})$ as the vector $y\in\mathbb{R}^{\mathcal{P}_{d_{1}+d_{2}-2k}}$ such that

[TABLE]

where the sum is over all $k$-matching $d_{1}$-precolorings $\psi_{1}$ and $d_{2}$-precolorings $\psi_{2}$. For near-cubic graphs $\tilde{G}_{1}=(G_{1},v_{1},\nu_{1})$ with $\deg(v_{1})=d_{1}$ and $\tilde{G}_{2}=(G_{2},v_{2},\nu_{2})$ with $\deg(v_{2})=d_{2}$, let $\gamma_{k}(\tilde{G}_{1},\tilde{G}_{2})$ denote the near-cubic graph $(G,v,\nu)$, where $G$ is obtained from $G_{1}$ and $G_{2}$ by identifying $v_{1}$ with $v_{2}$ to a single vertex $v$ and for $i\in\{0,1,\ldots,k-1\}$ removing the half-edges $\nu_{1}^{-1}(d_{1}-k+i)$ and $\nu_{2}^{-1}(d_{2}-1-i)$ and connecting the other halfs of the edges; and $\nu^{-1}(i)=\nu_{1}^{-1}(i)$ for $i\in\{0,\ldots,d_{1}-k-1\}$ and $\nu^{-1}(i)=\nu^{-1}_{2}(i-(d_{1}-k))$ for $i\in\{d_{1}-k,\ldots,d_{1}+d_{2}-2k-1\}$. See Figure 7 for an illustration.

Observation 11**.**

Let $\tilde{G}_{1}$ and $\tilde{G}_{2}$ be near-cubic graphs. For every integer $k\in\{0,\ldots,\min(d(\tilde{G}_{1}),d(\tilde{G}_{2}))\}$, we have $n_{\gamma_{k}(\tilde{G}_{1},\tilde{G}_{2})}=\gamma_{k}(n_{\tilde{G}_{1}},n_{\tilde{G}_{2}})$.

By a computer-assisted enumeration, we verified the following claim.

Lemma 12**.**

There exists cones $K_{d}\subseteq\mathbb{R}^{\mathcal{P}_{d}}$ for $d=2,\ldots,8$ such that the following claims hold.

• (a)

$K_{d}=B_{d}$* when $d\leq 4$ and $K_{5}=B^{\prime}_{5}$.*

• (b)

For all $d\in\{2,\ldots,8\}$, the elements of $K_{d}$ are invariant with respect to permutation of colors.

• (c)

For $d\in\{2,\ldots,7\}$, the cone $K_{d}$ is closed under rotations and flips.

• (d)

If $2\leq d_{1}\leq d_{2}$ and $d_{1}+d_{2}\leq 7$, then for all $x_{1}\in K_{d_{1}}$ and $x_{2}\in K_{d_{2}}$ we have $\gamma_{0}(x_{1},x_{2})\in K_{d_{1}+d_{2}}$.

• (e)

If $2\leq d\leq 5$, then for all $x\in K_{d}$ we have $\gamma_{1}(n_{\tilde{R}_{3,1}},x)\in K_{d+1}$.

• (f)

If $3\leq d\leq 7$, then for all $x\in K_{d}$ we have $\gamma_{2}(n_{\tilde{R}_{3,1}},x)\in K_{d-1}$.

• (g)

If $2\leq d_{1}\leq 6$ and $1\leq c\leq d_{1}/2$, then for all $x_{1}\in K_{d_{1}}$ and $x_{2}\in K_{7+2c-d_{1}}$, we have $\gamma_{c}(x_{1},x_{2})\in K_{7}$.

• (h)

For every $x_{1}\in K_{8}$ and $x_{2}\in K_{7}$, we have $\gamma_{4}(x_{1},x_{2})\in K_{7}$.

• (i)

For every $x_{1},x_{2}\in K_{6}$, we have $r_{2}(\gamma_{2}(x_{1},x_{2}))\in K_{8}$.

Proof.

The proof and the program to verify the proof can be found at http://lidicky.name/pub/4cone/. The cones are described by their rays, enumerated in the file. Cone $K_{6}$ has 102 rays, $K_{7}$ has 22605 rays, and $K_{8}$ has 4330 rays. It suffices to verify all the claims for $x$, $x_{1}$, $x_{2}$ being the rays of the cones specified in the claims; the inclusion of the resulting vectors in the appropriate cone is certified by expressing them as a linear non-negative combination of the rays of the cone. ∎

Parts (e) and (f) of Lemma 12 have the following corollary.

Lemma 13**.**

Let $\tilde{G}=(G,v,\nu)$ be a plane near-cubic graph and let $d=d(\tilde{G})$. If $d\in\{2,\ldots,7\}$ and $n_{\tilde{G}}\not\in K_{d}$, then there exists a plane near-cubic graph $\tilde{G}_{0}=(G_{0},v_{0},\nu_{0})$ such that $d(\tilde{G}_{0})=7$, $n_{\tilde{G}_{0}}\not\in K_{7}$, $G_{0}-v_{0}$ is an induced subgraph of $G-v$, and $|V(G_{0})|\leq|V(G)|-(7-d)$.

Proof.

We prove the claim by induction on the number of vertices of $G$. When $d\leq 4$, the claim is vacuously true by Theorem 5, since $K_{d}=B_{d}$. When $d=7$, we can set $\tilde{G}_{0}=\tilde{G}$. Hence, suppose that $d\in\{5,6\}$. Since $n_{\tilde{G}}\not\in K_{d}$, the function $n_{\tilde{G}}$ is not identically zero.

If $G-v$ is disconnected, we can by symmetry assume that $\tilde{G}=\gamma_{0}(\tilde{G}_{1},\tilde{G}_{2})$ for plane near-cubic graphs $\tilde{G}_{1}$ and $\tilde{G}_{2}$ such that $d=d(\tilde{G}_{1})+d(\tilde{G}_{2})$ and $d(\tilde{G}_{1})\leq d(\tilde{G}_{2})$. Since $n_{\tilde{G}}$ is not the zero function, $n_{\tilde{G}_{1}}$ is not the zero function either, and thus $d(\tilde{G}_{1})\neq 1$. Hence $d(\tilde{G}_{1})\geq 2$, and thus $2\leq d(\tilde{G}_{2})\leq 4$. Hence by Lemma 12(a), $n_{\tilde{G}_{1}}\in K_{d(\tilde{G}_{1})}$ and $n_{\tilde{G}_{2}}\in K_{d(\tilde{G}_{2})}$, and $n_{\tilde{G}}\in K_{d}$ by Lemma 12(d), which is a contradiction.

Hence, $G-v$ is connected (and the same argument as for disconnected $G-v$ shows that no loop is incident with $v$). Consequently, $v$ is not incident with a triple edge. If $v$ is incident with a double edge, then we can by symmetry assume that $\tilde{G}=\gamma_{1}(\tilde{R}_{3,1},\tilde{G}_{1})$ for a plane near-cubic graph $\tilde{G}_{1}=(G_{1},v_{1},\nu_{1})$ with $d(\tilde{G}_{1})=d-1\leq 5$. By Lemma 12(e), since $n_{\tilde{G}}\not\in K_{d}$, we have $n_{\tilde{G}_{1}}\not\in K_{d-1}$. By the induction hypothesis, there exists a plane near-cubic graph $\tilde{G}_{0}=(G_{0},v_{0},\nu_{0})$ with $d(\tilde{G}_{0})=7$, such that $n_{\tilde{G}_{0}}\not\in K_{7}$, $G_{0}-v_{0}$ is an induced subgraph of $G_{1}-v_{1}$, and thus also of $G-v$, and $|V(G_{0})|\leq|V(G_{1})|-(7-(d-1))<|V(G)|-(7-d)$, as required.

Hence, we can assume $v$ is not incident with a double edge. Consequently, we can by symmetry assume that $\tilde{G}=\gamma_{2}(\tilde{R}_{3,1},\tilde{G}_{1})$ for a plane near-cubic graph $\tilde{G}_{1}=(G_{1},v_{1},\nu_{1})$ with $d(\tilde{G}_{1})=d+1$. By Lemma 12(f), since $n_{\tilde{G}}\not\in K_{d}$, we have $n_{\tilde{G}_{1}}\not\in K_{d+1}$. By the induction hypothesis, there exists a plane near-cubic graph $\tilde{G}_{0}=(G_{0},v_{0},\nu_{0})$ with $d(\tilde{G}_{0})=7$, such that $n_{\tilde{G}_{0}}\not\in K_{7}$, $G_{0}-v_{0}$ is an induced subgraph of $G_{1}-v_{1}$, and $|V(G_{0})|\leq|V(G_{1})|-(7-(d+1))=|V(G)|-(7-d)$. Hence, the claim of the lemma follows. ∎

We will say that a plane near-cubic graph $\tilde{G}=(G,v,\nu)$ is extremal if $d(\tilde{G})=7$, $n_{\tilde{G}}\not\in K_{7}$, and there does not exist any plane near-cubic graph $\tilde{G}_{0}=(G_{0},v_{0},\nu)$ with $d(\tilde{G}_{0})=7$ such that $n_{\tilde{G}_{0}}\not\in K_{7}$ and $G_{0}-v_{0}$ is a proper minor of $G-v$.

Lemma 14**.**

If $\tilde{G}=(G,v,\nu)$ is an extremal plane near-cubic graph and $\tilde{G}^{\prime}=(G^{\prime},v^{\prime},\nu^{\prime})$ is a plane near-cubic graph with $d(\tilde{G}^{\prime})\leq 7$ such that $G^{\prime}-v^{\prime}$ is a proper minor of $G-v$, then $n_{\tilde{G}^{\prime}}\in K_{d(\tilde{G}^{\prime})}$.

Proof.

If $n_{\tilde{G}^{\prime}}\not\in K_{d(\tilde{G}^{\prime})}$, then by Lemma 13 there would exist a plane near-cubic graph $\tilde{G}_{0}=(G_{0},v_{0},\nu_{0})$ such that $d(\tilde{G}_{0})=7$, $n_{\tilde{G}_{0}}\not\in K_{7}$ and $G_{0}-v_{0}$ is an induced subgraph of $G^{\prime}-v^{\prime}$. However, then $G_{0}-v_{0}$ would be a proper minor of $G-v$, contradicting the assumption that $\tilde{G}$ is extremal. ∎

Next, let us explore consequences of part (g) of Lemma 12.

Lemma 15**.**

If $\tilde{G}=(G,v,\nu)$ is an extremal plane near-cubic graph, then $v$ is not incident with loops or parallel edges and $G-v$ is $2$-edge-connected.

Proof.

Analogously to the proof of Lemma 13, if $v$ were incident with a loop or a parallel edge or if $G-v$ were not $2$-edge-connected, we would have $\tilde{G}=\gamma_{c}(\tilde{G}_{1},\tilde{G}_{2})$ for plane near-cubic graphs $\tilde{G}_{1}$ and $\tilde{G}_{2}$ such that $2\leq d(\tilde{G}_{1})\leq d(\tilde{G}_{2})$, $d(\tilde{G}_{1})+d(\tilde{G}_{2})=7+2c$, and $c\leq 1$; in particular, $d(\tilde{G}_{2})\leq 7$ and $d(\tilde{G}_{1})\leq\lfloor(7+2c)/2\rfloor\leq 4$. By Lemma 14, we have $n_{\tilde{G}_{i}}\in K_{d(\tilde{G}_{i})}$ for $i\in\{1,2\}$. By Lemma 12(g), we conclude $n_{\tilde{G}}\in K_{7}$, which is a contradiction. ∎

Suppose $A$ and $B$ form a partition of the vertex set of a graph $H$, and let $S$ be the set of edges of $H$ with one end in $A$ and the other end in $B$. In this situation, we say $S$ is an edge cut of $H$ with sides $A$ and $B$.

Lemma 16**.**

If $\tilde{G}=(G,v,\nu)$ is an extremal plane near-cubic graph, then $G-v$ does not contain an edge cut $S$ such that $v$ has at least $|S|$ neighbors in each side of the cut.

Proof.

Suppose for a contradiction $G-v$ contains such an edge cut $S$ of size $c$, and thus $\tilde{G}=\gamma_{c}(\tilde{G}_{1},\tilde{G}_{2})$ for plane near-cubic graphs $\tilde{G}_{1}$ and $\tilde{G}_{2}$ such that $2c\leq d(\tilde{G}_{1})\leq d(\tilde{G}_{2})$ and $d(\tilde{G}_{1})+d(\tilde{G}_{2})=7+2c$. Since $v$ has $7$ neighbors and at least $c$ of them are contained in each of the sides of the cut, we have $c\leq 3$. Note that $d(\tilde{G}_{2})\leq 7$ and $d(\tilde{G}_{1})\leq\lfloor(7+2c)/2\rfloor\leq 6$. By Lemma 14, we have $n_{\tilde{G}_{i}}\in K_{d(\tilde{G}_{i})}$ for $i\in\{1,2\}$. By Lemma 12(g), we conclude $n_{\tilde{G}}\in K_{7}$, which is a contradiction. ∎

An edge cut $S$ of size at most five in a near-cubic graph $\tilde{G}=(G,v,\nu)$ is essential if the side of $S$ containing $v$ contains at least one other vertex and the other side $B$ of $S$ induces neither a tree nor a $5$-cycle.

Lemma 17**.**

If $\tilde{G}=(G,v,\nu)$ is an extremal plane near-cubic graph, then $\tilde{G}$ does not contain an essential edge cut $S$ of size at most five.

Proof.

Suppose for a contradiction that $\tilde{G}$ contains an essential edge-cut $S$ of size $k\leq 5$, and choose one with minimum $k$, and subject to that one for which the side $B$ not containing $v$ is minimal. We claim $G[B]$ is $2$-edge-connected. Otherwise, $B$ is a disjoint union of non-empty sets $B_{1}$ and $B_{2}$, where $G$ contains $r\leq 1$ edges with one end in $B_{1}$ and the other end in $B_{2}$. For $i\in\{1,2\}$, let $S_{i}$ denote the set of edges of $G$ with exactly one end in $B_{i}$. Since $\tilde{G}$ is extremal, $n_{\tilde{G}}\not\in K_{7}$ is not identically zero, and thus $G$ is $2$-edge-connected, implying $|S_{i}|\geq 2$. Hence, $|S_{i}|=k+2r-|S_{3-i}|\leq k$. By the minimality of $B$, we conclude that $B_{i}$ induces a tree or a $5$-cycle, and thus $|S_{i}|\geq 3$. Hence $5\geq k=|S_{1}|+|S_{2}|-2r\geq 6-2r$, and thus $r=1$ and $|S_{1}|,|S_{2}|\leq 4$. This implies that neither $B_{1}$ nor $B_{2}$ induces a $5$-cycle, and thus both of them induce trees; and $G$ contains an edge between them, implying that $B$ induces a tree, contrary to the assumption that $S$ is an essential edge cut.

Since $G[B]$ is $2$-edge-connected and subcubic, each face of $G[B]$ is bounded by a cycle. Let $C_{S}$ denote the cycle bounding the face $f$ of $G[B]$ whose interior contains $v$. Observe that all edges of $S$ are drawn inside $f$. Otherwise, the set $S^{\prime}$ of edges of $S$ drawn inside $C$ forms an edge cut of order smaller than $k$ and by the minimality of $k$, its side $B^{\prime}\supsetneq B$ induces a tree or a $5$-cycle; this is not possible, since $G[B]$ is $2$-edge connected and not a tree.

Let $\tilde{G}_{c}$ be the plane near-cubic graph obtained from $G$ by contracting the side of the cut containing $v$ to a single vertex. By Lemma 14, we have $n_{\tilde{G}_{c}}\in K_{k}$. Since $K_{d}=B_{d}$ for $d\leq 4$ and $K_{5}=B^{\prime}_{5}$,

[TABLE]

where $i\leq 11$ if $k=5$ and the coefficients $c_{i}$ are non-negative. Let $\tilde{G}_{i}=(G_{i},v_{i},\nu_{i})$ denote the plane near-cubic graph obtained from $\tilde{G}$ by replacing the side of the cut $S$ not containing $v$ by $\tilde{R}_{k,i}$. Note that $n_{\tilde{G}}=\sum_{i}c_{i}n_{\tilde{G}_{i}}$, and since $K_{7}$ is a cone and $n_{\tilde{G}}\not\in K_{7}$, there exists $i$ such that $n_{\tilde{G}_{i}}\not\in K_{7}$. Because $B$ contains the cycle $C_{S}$ and all edges of $S$ are incident with vertices of $C_{S}$, we see $G_{i}-v_{i}$ is a proper minor of $G-v$, contradicting the extremality of $\tilde{G}$. ∎

In Lemma 15, we argued that if $\tilde{G}=(G,v,\nu)$ is an extremal plane near-cubic graph, then $G-v$ is $2$-edge-connected, and thus its face containing $v$ is bounded by a cycle $C$. Let us now argue that the graph stays $2$-edge-connected after removing $V(C)$ as well.

Lemma 18**.**

Let $\tilde{G}=(G,v,\nu)$ be an extremal plane near-cubic graph and let $C$ be the cycle bounding the face of $G-v$ containing $v$. The cycle $C$ is induced, no two neighbors of $v$ in $C$ are adjacent, and the graph $G-(V(C)\cup\{v\})$ is $2$-edge-connected and has more than one vertex.

Proof.

Consider a simple closed curve $c$ in the plane intersecting $G$ in two edges of $C$, $b\leq 4$ edges incident with $v$, and $r\leq 1$ edges of $E(G-v)\setminus E(C)$, where each edge is intersected at most once. The curve $c$ separates the plane into two parts; let $A$ and $B$ be the corresponding partition of vertices of $G$, where $v\in A$, and let $S$ be the edge cut in $G$ consisting of the edges with one end in $A$ and the other end in $B$. By Lemma 16 applied to the edge cut in $G-v$ obtained from $S$ by removing the edges incident with $v$, it follows that $b\leq r+1$, and thus $|S|\leq 3+2r\leq 5$. By Lemma 17 we conclude that the edge cut satisfies one of the following conditions.

• •

$r=0$, $b=1$, $|S|=3$, and $B$ consists of a single vertex of $C$, or

• •

$r=1$ and $G[B]$ is a subpath of $C$, or

• •

$r=1$, $b=2$, and $G[B]$ is a $5$-cycle containing exactly one vertex not in $V(C)$.

If $C$ had a chord $e$, this would give a contradiction by considering a curve $c$ (with $r=0$) drawn next to the chord so that $e\in E(G[B])$ and $b\leq 3$; hence, $C$ is an induced cycle. If two neighbors of $v$ in $C$ were adjacent, we would obtain a contradiction by considering a curve $c$ (with $r=0$ and $b=2$) drawn around them. If the graph $G-(V(C)\cup\{v\})$ were not connected, we would obtain a contradiction by considering a curve $c$ (with $r=0$ and $b\leq 3$) chosen so that both $A$ and $B$ contain a vertex of $G-(V(C)\cup\{v\})$. Finally, if the graph $G-(V(C)\cup\{v\})$ were not $2$-edge-connected, then we could choose $c$ so that $r=1$, $b\leq 3$, and $B$ contains a vertex of $G-(V(C)\cup\{v\})$. But then $G[B]$ would be a $5$-cycle containing exactly one vertex not in $V(C)$ and consequently two adjacent vertices of $C$ would be neighbors of $v$, which is a contradiction.

Therefore, the graph $G-(V(C)\cup\{v\})$ is $2$-edge-connected. Since no two neighbors of $v$ in $C$ are adjacent, $G$ contains at least $7$ edges between $V(C)$ and $V(G)\setminus(V(C)\cup\{v\})$, and thus $G-(V(C)\cup\{v\})$ has more than one vertex. ∎

Finally, let us apply the parts (h) and (i) of Lemma 12.

Lemma 19**.**

If $\tilde{G}=(G,v,\nu)$ is an extremal plane near-cubic graph, then $G$ has at least $28$ vertices.

Proof.

Recall that by the definition of extremal, $d(\tilde{G})=7$. By Lemma 15, the face of $G-v$ containing $v$ is bounded by a cycle $C$. Let $v_{1}$, …, $v_{7}$ be the neighbors of $v$ in $C$ in order. For $i\in\{1,\ldots,7\}$, let $P_{i}$ denote the subpath of $C$ from $v_{i}$ to $v_{i+1}$ (where $v_{8}=v_{1}$).

By Lemma 18, the cycle $C$ is induced, no two neighbors of $v$ in $C$ are adjacent, and the graph $G-(V(C)\cup\{v\})$ is $2$-edge-connected and has more than one vertex. Hence, the face of $G-(V(C)\cup\{v\})$ containing $v$ is bounded by a cycle $C^{\prime}$. For a subgraph $G^{\prime}\subseteq G$ containing $C\cup C^{\prime}$, let $X(G^{\prime})$ denote the set of faces of $G^{\prime}$ separated from $v$ by $C^{\prime}$ and let $Y(G^{\prime})$ denote the set of faces of $G^{\prime}$ separated from $v$ by $C$ but not by $C^{\prime}$. See Figure 8(a) for an example. For $i\in\{1,\ldots,7\}$, we say that a face $f\in X(G^{\prime})$ sees $P_{i}$ if there exists a face $f^{\prime}\in Y(G^{\prime})$ such that $f^{\prime}$ is incident with an edge of $P_{i}$ and the boundaries of $f$ and $f^{\prime}$ share at least one edge.

If for some $i\in\{1,\ldots,7\}$, some face of $X(G)$ saw $P_{i}$, $P_{i+2}$, and $P_{i+4}$ (with indices taken cyclically) then $\tilde{G}=\gamma_{4}(r_{2}(\gamma_{2}(\tilde{G}_{1},\tilde{G}_{2})),\tilde{G}_{3})$ for plane near-cubic graphs $\tilde{G}_{1}$, $\tilde{G}_{2}$, and $\tilde{G}_{3}$ with $d(\tilde{G}_{1})=d(\tilde{G}_{2})=6$ and $d(\tilde{G}_{3})=7$ (see Figure 8(b)). Lemma 14 would imply $n_{\tilde{G}_{j}}\in K_{d(\tilde{G}_{j})}$ for $j\in\{1,2,3\}$, and by Lemma 12(h) and (i), we would have $n_{\tilde{G}}\in K_{7}$, which is a contradiction. Hence,

[TABLE]

Let $b_{1}$ be the number of edges of $G$ with one end in $C$ and the other end in $C^{\prime}$, let $b_{2}$ be the number of chords of $C^{\prime}$, let $b_{3}$ be the number of edges with one end in $C^{\prime}$ and the other end in $V(G)\setminus V(C\cup C^{\prime})$, and let $b_{4}$ be the number of edges of $G-v-V(C\cup C^{\prime})$. Note that $b_{1}\geq 7$, $b_{3}$ is at least three times the number of components of $G-v-V(C\cup C^{\prime})$, $|E(C)|=7+b_{1}$, $|E(C^{\prime})|=b_{1}+2b_{2}+b_{3}$, and

[TABLE]

A case analysis shows that since (3) holds, one of the following conditions holds:

• •

$b_{1}\geq 8$ and $b_{2}\geq 2$, or

• •

$b_{1}\geq 8$ and $b_{3}\geq 3$, or

• •

$b_{3}\geq 6$, or

• •

$b_{3}\geq 4$ and $b_{4}\geq 1$.

Hence $3b_{1}+3b_{2}+2b_{3}+b_{4}\geq 30$, and thus $G$ has at least $44$ edges. Consequently, $|V(G)|\geq(2|E(G)|-4)/3\geq 28$. ∎

As a consequence, this verifies Conjecture 8 for small graphs.

Corollary 20**.**

Conjecture 8 holds for all plane near-cubic graphs with less than $30$ vertices.

Proof.

Let $\tilde{G}=(G,v,\nu)$ be a counterexample to Conjecture 8, and in particular $n_{\tilde{G}}\not\in B^{\prime}_{5}=K_{5}$. By Lemma 13, there exists a plane near-cubic graph $\tilde{G}_{0}=(G_{0},v_{0},\nu_{0})$ such that $d(\tilde{G}_{0})=7$, $n_{\tilde{G}_{0}}\not\in K_{7}$, and $|V(G_{0})|\leq|V(G)|-2$. Hence, there exists an extremal plane near-cubic graph $\tilde{G}_{1}=(G_{1},v_{1},\nu_{1})$ such that $|V(G_{1})|\leq|V(G_{0})|$. By Lemma 19, we have $|V(G_{1})|\geq 28$, and thus $|V(G)|\geq 30$. ∎

Note that the analysis at the end of the proof of Lemma 19 can be improved. By a computer-assisted enumeration, one can show that to ensure that (3) holds, $G-v$ must contain one of 38 specific graphs as a minor; the smallest are depicted in Figure 9. Hence, every counterexample to Conjecture 8 must contain one of these 38 as a minor. The list of these 38 graphs is available at http://lidicky.name/pub/4cone/.