(4, 2)-Choosability of Planar Graphs with Forbidden Structures

Abstract

All planar graphs are 4-colorable and 5-choosable, while some planar graphs are not 4-choosable. Determining which properties guarantee that a planar graph can be colored using lists of size four has received significant attention. In terms of constraining the structure of the graph, for any $$\ell \in \{3,4,5,6,7\}$$ , a planar graph is 4-choosable if it is $$\ell $$ -cycle-free. In terms of constraining the list assignment, one refinement of k-choosability is choosability with separation. A graph is (k, s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists. Every planar graph is (4, 1)-choosable, but there exist planar graphs that are not (4, 3)-choosable. It is an open question whether planar graphs are always (4, 2)-choosable. A chorded $$\ell $$ -cycle is an $$\ell $$ -cycle with one additional edge. We demonstrate for each $$\ell \in \{5,6,7\}$$ that a planar graph is (4, 2)-choosable if it does not contain chorded $$\ell $$ -cycles.