The edge colorings of $K_5$-minor free graphs

Abstract

In 1965, Vizing proved that every planar graph $G$ with maximum degree $\Delta \ge 8$ is edge $\Delta$-colorable. It is also proved that every planar graph $G$ with maximum degree $\Delta =7$ is edge $\Delta$-colorable by Sanders and Zhao, independently by Zhang. In this paper, we extend the above results by showing that every $K_5$-minor free graph with maximum degree $\Delta$ at least seven is edge $\Delta$-colorable.

Introduction

All graphs considered in this paper are finite, undirected and simple. Let $G$ be a graph. We use $V\left(G\right),E\left(G\right),\Delta \left(G\right)$ and $\delta \left(G\right)$ (or simply $V,E,\Delta$ and $\delta$) to denote the vertex set, the edge set, the maximum degree and the minimum degree of $G$, respectively. For a vertex $v\in V\left(G\right)$, let $N_G\left(v\right)=\{u\in V\left(G\right)\mid uv\in E\left(G\right)\}$ be the set of neighbors of $v$. Furthermore, let $N_G\left(X\right)={\bigcup }_{u\in X}{N}_G\left(u\right)\setminus X$ for a subset $X\subseteq V\left(G\right)$. A $k$-cycle is a cycle of length $k$. A 3-cycle is also said to be a triangle.

An edge $k$-coloring of a graph $G$ is an assignment of $k$ colors $1,2,\dots ,k$ to the edges of $G$ such that no two adjacent edges receive the same color. The minimum integer $k$ such that $G$ admits an edge $k$-coloring is called the chromatic index of $G$ and is denoted by ${\chi }^{\prime }\left(G\right)$. For any graph $G$, it is obvious that ${\chi }^{\prime }\left(G\right)\ge \Delta \left(G\right)$. Vizing [6] and Gupta [1] independently proved that ${\chi }^{\prime }\left(G\right)\le \Delta \left(G\right)+1$. This leads to a natural classification of graphs into two classes. A graph is said to be class 1 if ${\chi }^{\prime }\left(G\right)=\Delta \left(G\right)$ and of class 2 if ${\chi }^{\prime }\left(G\right)=\Delta \left(G\right)+1$.

The problem of deciding whether a graph is class 1 or class 2 is NP-hard, see Holyer [2]. It is reasonable to consider the problem for some special classes of graphs, such as planar graphs. In [8], Vizing gave some examples of planar graphs with maximum degree at most five which are of class 2. He also proved that any planar graph with maximum degree at least eight is of class 1 [7]. Planar graphs with maximum degree seven are of class 1 are proved by Sanders and Zhao [5], independently by Zhang [10]. It remains an open problem that any planar graph with maximum degree six is of class 1. This problem is affirmative provided some additional conditions, see [11].

By contracting an edge $e$ of a graph $G$, we mean that deleting $e$ from $G$ and then identifying its end-vertices and deleting all multiple edges. A graph $H$ is a minor of a graph $G$ if $H$ can be obtained from $G$ by deleting edges, deleting vertices and contracting edges. A graph $G$ is called $H$-minor free if $G$ has no minor which is isomorphic to $H$. It is well-known that every planar graph contains neither $K_5$-minor nor $K_{3,3}$-minor. Therefore, the family of $K_5$-minor free graphs is a generalization of planar graphs. The goal of this paper is to extend the result from planar graphs in [5], [10] to $K_5$-minor free graphs. The main result of this paper is as follows.

Theorem 1.1

Let $G$ be a $K_5$-minor free graph with maximum degree $\Delta \left(G\right)\ge 7$. Then ${\chi }^{\prime }\left(G\right)=\Delta \left(G\right)$.

The remainder of this paper is organized as follows. In Section 2, we prove several properties of $K_5$-minor free graphs; and in Section 3, we prove Theorem 1.1 based on the results in Section 2.