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

TL;DR

This paper characterizes a class of simple graphs with limited odd cycle intersections and proves they satisfy the list-edge-coloring conjecture, including stronger results for graphs with maximum degree at least 4.

Contribution

It provides a structural characterization of graphs with odd cycles intersecting in at most one edge and proves they satisfy the list-edge-coloring conjecture, with enhanced results for higher degree graphs.

Findings

Graphs in the class satisfy the list-edge-coloring conjecture.

For graphs with maximum degree at least 4, they are $(m riangle(G):m)$-edge-choosable.

Graphs with $ riangle(G) eq 3$ are $ riangle(G)$-edge-paintable.

Abstract

We study the class of simple graphs $\mathcal{G}^*$ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in $\mathcal{G}^*$ and prove that every $G \in \mathcal{G}^*$ satisfies the list-edge-coloring conjecture. When $\Delta(G) \geq 4$, 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 $m \geq 1$.