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

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