(4,2)-choosability of planar graphs with forbidden structures

TL;DR

This paper investigates the (4,2)-choosability of planar graphs, establishing conditions under which such graphs are (4,2)-choosable based on forbidden chorded cycles of certain lengths.

Contribution

The paper proves that planar graphs without chorded 5-, 6-, or 7-cycles are (4,2)-choosable, advancing understanding of list coloring under structural constraints.

Findings

Planar graphs without chorded 5-cycles are (4,2)-choosable.

Planar graphs without chorded 6-cycles are (4,2)-choosable.

Planar graphs without chorded 7-cycles are (4,2)-choosable.

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.