The list chromatic index of simple graphs whose odd cycles intersect in at most one edge

Abstract

We study the class of simple graphs ${\mathcal{G}}^{\ast }$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in ${\mathcal{G}}^{\ast }$ and prove that every $G\in {\mathcal{G}}^{\ast }$ satisfies the list-edge-coloring conjecture. When $\Delta (G)\ge 4$, we in fact prove a stronger result about kernel-perfect orientations in $L(G)$ which implies that $G$ is $(m\Delta (G):m)$-edge-choosable and $\Delta (G)$-edge-paintable for every $m\ge 1$.

15 pages, 8 figures. Minor fixes throughout the paper