On incidence choosability of cubic graphs

Abstract

An incidence of a graph G is a pair (u, e) where u is a vertex of G and e is an edge of G incident with u. Two incidences (u, e) and (v, f ) of G are adjacent whenever (i) u = v, or (ii) e = f , or (iii) uv = e or uv = f . An incidence k-coloring of G is a mapping from the set of incidences of G to a set of k colors such that every two adjacent incidences receive distinct colors. The notion of incidence coloring has been introduced by Brualdi and Quinn Massey (1993) from a relation to strong edge coloring, and since then, attracted by many authors.On a list version of incidence coloring, it was shown by Benmedjdoub et. al. (2017) that every Hamiltonian cubic graph is incidence 6-choosable. In this paper, we show that every cubic (loopless) multigraph is incidence 6-choosable. As a direct consequence, it implies that the list strong chromatic index of a (2, 3)-bipartite graph is at most 6, where a (2,3)-bipartite graph is a bipartite graph such that one partite set has maximum degree at most 2 and the other partite set has maximum degree at most 3. *

(2, 3)-bipartite graphs are strongly 6-edge-choosable

“…Note that no counterexample for list-coloring version of Conjecture 1 is known. The aim of this note is to show the following result (already proved by Kang and Park in [6] using different techniques):…”