Choosability with union separation of planar graphs without cycles of length 4

Abstract

Abstract For a graph G and a positive integer k, a k-list assignment of G is a function L on the vertices of G such that for each vertex v ∈ V(G), |L(v)| ≥ k. Let s be a nonnegative integer. Then L is a ( k , k + s ) -list assignment of G if | L ( u ) ∪ L ( v ) | ≥ k + s for each edge uv. If for each ( k , k + s ) -list assignment L of G, G admits a proper coloring φ such that φ(v) ∈ L(v) for each v ∈ V(G), then we say G is ( k , k + s ) -choosable. This refinement of choosability is called choosability with union separation by Kumbhat, Moss and Stolee, who showed that all planar graphs are (3, 11)-choosable. In this paper, we prove that every planar graph without cycles of length 4 is (3,6)-choosable.