On Choosability with Separation of Planar Graphs Without Adjacent Short Cycles

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 prove that planar graphs without 4-cycles adjacent to $$4^-$$ -cycles are (3, 1)-choosable. This is a strengthening of a result which says that planar graphs without 4-cycles are (3, 1)-choosable.