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

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 nu is an element of V(G), vertical bar L(nu)vertical bar >= k. Let s be a nonnegative integer. Then L is a (k, k + s)-list assignment of G if vertical bar L(u) boolean OR L(nu)vertical bar >= k + s for each edge uv. If for each (k, k s)-list assignment L of G, G admits a proper coloring phi such that phi(nu) is an element of L(nu) for each nu is an element of 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. (C) 2020 Elsevier Inc. All rights reserved.