A sufficient condition for planar graphs to be (3, 1)-choosable

Abstract

A (k, d)-list assignment L of a graph is a function that assigns to each vertex v a list L(v) of at least k colors satisfying $|L(x)\cap L(y)|\le d$ for each edge xy. An L-coloring is a vertex coloring $\pi$ such that $\pi (v) \in L(v)$ for each vertex v and $\pi (x) \ne \pi (y)$ for each edge xy. A graph G is (k, d)-choosable if there exists an L-coloring of G for every (k, d)-list assignment L. This concept is known as choosability with separation. In this paper, we will use Thomassen list coloring extension method to prove that planar graphs with neither 6-cycles nor adjacent 4- and 5-cycles are (3, 1)-choosable. This is a strengthening of a result obtained by using Discharging method which says that planar graphs without 5- and 6-cycles are (3, 1)-choosable.