The Chromatic Index of a Claw-Free Graph Whose Core has Maximum Degree $2$

Abstract

Let $G$ be a graph. The core of $G$, denoted by $G_{\Delta }$, is the subgraph of $G$ induced by the vertices of degree $\Delta (G)$, where $\Delta (G)$ denotes the maximum degree of $G$. A $k$-edge coloring of $G$ is a function $f:E(G)\rightarrow L$ such that $|L| = k$ and $f(e_1)\ne f(e_2)$, for any two adjacent edges $e_1$ and $e_2$ of $G$. The chromatic index of $G$, denoted by $\chi '(G)$, is the minimum number $k$ for which $G$ has a $k$-edge coloring. A graph $G$ is said to be Class $1$ if $\chi '(G) = \Delta (G)$ and Class $2$ if $\chi '(G) = \Delta (G) + 1$. Hilton and Zhao conjectured that if $G$ is a connected graph, $\Delta (G_{\Delta })\le 2$, and $G$ is not the Petersen graph with one vertex removed, then $G$ is Class $2$ if and only if $G$ is overfull. In this paper, we prove this conjecture for claw-free graphs of even order.