On the Edge-Colouring of Graphs whose Core has Maximum Degree Two

Abstract

The core $G_{\mathrm{\Delta }}$ is the subgraph of $G$ induced by the vertices of maximum degree. If $\mathrm{\Delta }(G)$ is a forest, then Fournier [8] showed that $G$ is Class 1. Therefore, if $G$ is Class 2, $G_{\mathrm{\Delta }}$ must contain cycles. A natural question is this: what is the chromatic index of $G$ if $G_{\mathrm{\Delta }}$ is a union of vertex-disjoint cycles? In this paper, we give further information about this question.