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 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.