On [formula omitted]-choosability of planar graphs without short cycles

Abstract

For a graph G and two positive integers t,r, a (t,t+r)-list assignment of G is a function L that assigns a set of permissible colors L(u) to every vertex u such that |L(u)|≥t and |L(u)∪L(w)|≥t+r when uw is an edge. The graph G is said to be (t,t+r)-choosable if G allows a proper coloring ψ satisfying ψ(u)∈L(u) for each u∈V(G) and each (t,t+r)-list assignment L of G. In this paper, we consider the (2,2+r)-choosability of planar graphs without short cycles. We show that: (1) if G is a planar graph contains no cycles of length 4, then G is (2,9)-choosable; (2) if G is a planar graph contains no cycles of lengths 4 and 5, then G is (2,7)-choosable.