Remarks on planar edge-chromatic critical graphs

TL;DR

This paper proves new structural properties of planar edge-chromatic critical graphs, providing insights that support Vizing's conjecture for planar graphs with maximum degree 6.

Contribution

It establishes the non-existence of certain 6-critical plane graphs with limited face incidences and characterizes face adjacencies in 5-critical plane graphs, advancing understanding of planar graph colorings.

Findings

No 6-critical plane graph with vertices incident to at most three 3-faces exists.

Every 5-critical plane graph has a 3-face adjacent to a 3- or 4-face.

Results support Vizing's conjecture for planar graphs with maximum degree 6.

Abstract

The only open case of Vizing's conjecture that every planar graph with $\Delta\geq 6$ is a class 1 graph is $\Delta = 6$. We give a short proof of the following statement: there is no 6-critical plane graph $G$, such that every vertex of $G$ is incident to at most three 3-faces. A stronger statement without restriction to critical graphs is stated in \cite{Wang_Xu_2013}. However, the proof given there works only for critical graphs. Furthermore, we show that every 5-critical plane graph has a 3-face which is adjacent to a $k$-face $(k\in \{3,4\})$. For $\Delta = 5$ our result gives insights into the structure of planar $5$-critical graphs, and the result for $\Delta=6$ gives support for the truth of Vizing's planar graph conjecture.

Full text

For $\Delta=5$ our result gives insights into the structure of planar $5$-critical graphs, and the result for $\Delta=6$ gives support for the truth of Vizing’s planar graph conjecture.

Keywords: planar graph; edge coloring; Vizing’s conjecture; critical graph

1 Introduction

We consider finite simple graphs $G$ with vertex set $V(G)$ and edge set $E(G)$. The vertex-degree of $v\in V(G)$ is denoted by $d_{G}(v)$, and $\Delta(G)$ denotes the maximum vertex-degree of $G$. If it is clear from the context, then $\Delta$ is frequently used. A graph is planar if it is embeddable into the Euclidean plane. A plane graph $(G,\Sigma)$ is a planar graph $G$ together with an embedding $\Sigma$ of $G$ into the Euclidean plane. If $(G,\Sigma)$ is a plane graph, then $F(G)$ denotes the set of faces of $(G,\Sigma)$. The degree $d_{(G,\Sigma)}(f)$ of a face $f$ is the length of its facial circuit. A face $f$ is a $k$-face if $d_{G}(f)=k$, and it is a $k^{+}$-face if $d_{G}(f)\geq k$.

The edge-chromatic number $\chi^{\prime}(G)$ of a graph $G$ is the minimum $k$ such that $G$ admits a proper $k$-edge-coloring. Vizing [4] proved that $\chi^{\prime}(G)\in\{\Delta(G),\Delta(G)+1\}$. If $\chi^{\prime}(G)=\Delta(G)$, then $G$ is a class 1 graph, and it is a class 2 graph otherwise. A class 2 graph $H$ is $k$-critical, if $\Delta(H)=k$ and $\chi^{\prime}(H^{\prime})<\chi^{\prime}(H)$ for every proper subgraph $H^{\prime}$ of $H$.

Vizing [4] showed for each $k\in\{2,3,4,5\}$ that there is a planar class 2 graph $G$ with $\Delta(G)=k$. He proved that every planar graph with $\Delta\geq 8$ is a class 1 graph, and conjectured that every planar graph with $\Delta\in\{6,7\}$ is a class 1 graph. Vizing’s conjecture is proved for planar graphs with $\Delta=7$ by Grünewald [1], Sanders, Zhao [3], and Zhang [7] independently. It is still open for the case $\Delta=6$. The paper provides short proofs for the following statements.

Theorem 1.1**.**

There is no 6-critical plane graph $(G,\Sigma)$, such that every vertex of $G$ is incident to at most three 3-faces.

If Vizing’s conjecture is not true, then every 6-critical graph has the following property.

Corollary 1.2**.**

Let $(G,\Sigma)$ be a plane graph. If $G$ is $6$-critical, then there is a vertex of $G$ which is incident to at least four $3$-faces.

Theorem 1.3**.**

Let $(G,\Sigma)$ be a plane graph. If $G$ is $5$-critical, then $(G,\Sigma)$ has a $3$-face which is adjacent to a $3$-face or to a $4$-face.

A significant longer proof of Theorem 1.1 is given in [5], but the statement is formulated for plane graphs. However, the proof works for critical graphs only. The assumption that a minimal counterexample is critical is wrong. It might be that a subgraph of this minimal counterexample $G$ does not fulfill the pre-condition of the statement. For example, if $G$ has a triangle $[vxyv]$ and a bivalent vertex $u$ such that $u$ is the unique vertex inside $[vxyv]$ and $u$ is adjacent to $x$ and $y$, then the removal of $u$ increases the number of 3-faces containing $v$ (see Figure 1).

2 Proofs of Theorems 1.1 and 1.3

We will use the following two lemmas.

Lemma 2.1** ([2]).**

If $G$ is a $6$-critical graph, then $|E(G)|\geq\frac{1}{2}(5|V(G)|+3)$.

Lemma 2.2** ([6]).**

If $G$ is a $5$-critical graph, then $|E(G)|\geq\frac{15}{7}|V(G)|$.

Proof of Theorem 1.1

Suppose to the contrary that there is a counterexample to the statement. Then there is a $6$-critical graph $G$ which has an embedding $\Sigma$ such that every $v\in V(G)$ is incident to at most three 3-faces. With Euler’s formula and Lemma 2.1 we deduce $\sum_{f\in F(G)}(d_{G}(f)-4)=2|E(G)|-4|F(G)|=2|E(G)|-4(|E(G)|+2-|V(G)|)\leq-|V(G)|-11$. Therefore, $|V(G)|+\sum_{f\in F(G)}(d_{G}(f)-4)\leq-11$.

Give initial charge 1 to each $v\in V(G)$ and $d_{G}(f)-4$ to each $f\in F(G)$. Discharge the elements of $V(G)\cup F(G)$ according to the following rule:

R1: Every vertex sends $\frac{1}{3}$ to its incident 3-faces.

The rule only moves the charge around and does not affect the sum. Furthermore, the finial charge of every vertex and face is at least 0. Therefore, $0\leq\sum_{v\in V(G)}1+\sum_{f\in F(G)}(d_{G}(f)-4)=|V(G)|+\sum_{f\in F(G)}(d_{G}(f)-4)\leq-11$, a contradiction.

Proof of Theorem 1.3

Suppose to the contrary that there is a counterexample to the statement. Then there is a $5$-critical graph $G$ which has an embedding $\Sigma$ such that every 3-face is adjacent to $5^{+}$-faces only. Hence, every vertex of $G$ is incident to at most two 3-faces, and every vertex which is incident to a 3-face is also incident to a $5^{+}$-face. By Lemma 2.2, we have $\sum_{f\in F(G)}(d_{G}(f)-4)\leq-\frac{2}{7}|V(G)|-8$. Therefore, $\frac{2}{7}|V(G)|+\sum_{f\in F(G)}(d_{G}(f)-4)\leq-8$.

Give initial charge of $\frac{2}{7}$ to each vertex and $d_{G}(f)-4$ to each face of $G$. Discharge the elements of $V(G)\cup F(G)$ according to the following rules:

R1: Every vertex sends $\frac{1}{3}$ to its incident 3-faces.

R2: Every $5^{+}$-face sends $\frac{d_{G}(f)-4}{d_{G}(f)}$ to its incident vertices.

Denote the finial charge by $ch^{*}$. Rules R1 and R2 imply that $ch^{*}(f)\geq 0$ for every $f\in F(G)$. Let $n\leq 2$ and $v$ be a vertex which is incident to $n$ 3-faces. If $n=0$, then $ch^{*}(v)\geq\frac{2}{7}>0$. If $n=1$, then $v$ is incident to at least one $5^{+}$-face, and therefore, $ch^{*}(v)\geq\frac{2}{7}+\frac{1}{5}-\frac{1}{3}>0$ by rule R2. If $n=2$, then $v$ is incident to at least two $5^{+}$-faces, and therefore $ch^{*}(v)\geq\frac{2}{7}+2\times\frac{1}{5}-2\times\frac{1}{3}=\frac{2}{105}>0$, by rule R2. Hence, $0\leq\sum_{v\in V(G)}\frac{2}{7}+\sum_{f\in F(G)}(d_{G}(f)-4)\leq-8$, a contradiction.