On (3, r)-Choosability of Some Planar Graphs

Abstract

There are many refinements of list coloring, one of which is the choosability with union separation. Let k, s be positive integers and let G be a graph. A $(k,k+s)$-list assignment of G is a mapping L assigning each vertex $v\in V(G)$ a list of colors L(v) such that $|L(v)|\ge k$ for each vertex $v\in V(G)$, and $|L(u)\cup L(v)|\ge k+s$ for each edge $uv\in E(G)$. If for each $(k,k+s)$-list assignment L of G, G admits a proper coloring $\varphi$ such that $\varphi (v)\in L(v)$ for each $v\in V(G)$, then G is $(k,k+s)$-choosable. Let G be a planar graph. In this paper, we prove: (1) if G contains no chorded 4-cycle, then G is (3, 8)-choosable; (2) if G contains neither intersecting triangles nor intersecting 4-cycles, then G is (3, 6)-choosable.