A Note on Edge Choosability and Degeneracy of Planar Graphs

Abstract

Let H be the graph obtained from a 6-cycle C ₆ by adding an edge which joins a pair of two vertices with distance two. We show that if a planar graph does not contain H, then G is edge-t-choosable, where t = 7 if Δ(G) = 5, and t = Δ(G) + 1, otherwise. This extends the known results that a planar graph is edge-(Δ(G) + 1)-choosable when Δ(G) ≠ 7 and G does not contain a k-cycle for some k ∈ {3, 5, 6}. It is well-known that {3, 5, 6} are only integers for which the lack of a cycle of length in {3, 5, 6} for a planar graph G implies 3-degeneracy of G. As a by-product, we prove that if a planar graph G contains at most seven 3-cycles, G is 3-degenerate. We also answer a problem of Raspaud and Wang (European J. Combin. 29(2008) 1064-1075) in negative.