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

We study the class of simple graphs G∗ for which every pair of distinct odd cycles intersect in at most one edge. We give a structural characterization of the graphs in G∗ and prove that every G∈G∗ satisfies the list-edge-coloring conjecture. When Δ(G)≥4, we in fact prove a stronger result about kernel-perfect orientations in L(G) which implies that G is (mΔ(G):m)-edge-choosable and Δ(G)-edge-paintable for every m≥1.