Planar graphs without cycles of lengths 4 and 5 and close triangles are DP-3-colorable

TL;DR

This paper proves new results on DP-3-colorability of planar graphs with forbidden cycles and specified distances between triangles, extending previous list-coloring bounds.

Contribution

It establishes that certain planar graphs without 4- and 5-cycles are DP-3-colorable when the distance between triangles exceeds specific thresholds, improving known bounds.

Findings

Planar graphs without 4,5-cycles and triangle distance ≥ 3 are DP-3-colorable.

Planar graphs without 4,5,6-cycles and triangle distance ≥ 2 are DP-3-colorable.

Corollary: bounds on $d_0$ and $d_1$ are improved to 3 and 2 respectively.

Abstract

Montassier, Raspaud, and Wang (2006) asked to find the smallest positive integers $d_0$ and $d_1$ such that planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge d_0$ are $3$-choosable and planar graphs without $\{4,5,6\}$-cycles and $d^{\Delta}\ge d_1$ are $3$-choosable, where $d^{\Delta}$ is the smallest distance between triangles. They showed that $2\le d_0\le 4$ and $d_1\le 3$. In this paper, we show that the following planar graphs are DP-3-colorable: (1) planar graphs without $\{4,5\}$-cycles and $d^{\Delta}\ge 3$ are DP-$3$-colorable, and (2) planar graphs without $\{4,5,6\}$-cycles and $d^{\Delta}\ge 2$ are DP-$3$-colorable. DP-coloring is a generalization of list-coloring, thus as a corollary, $d_0\le 3$ and $d_1\le 2$. We actually prove stronger statements that each pre-coloring on some cycles can be extended to the whole graph.