Choosability with union separation of triangle-free planar graphs

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 et al. (2018), who showed that all planar graphs are (3,11)-choosable and (4,9)-choosable. In this paper, we prove that every triangle-free planar graph is (3,7)-choosable. We also prove that every planar graph with girth at least 5 is (2,7)-choosable.