DP-4-coloring of planar graphs with some restrictions on cycles

TL;DR

This paper establishes three new sufficient conditions under which planar graphs with certain cycle restrictions are DP-4-colorable, expanding understanding of graph coloring in planar graphs.

Contribution

It provides three novel sufficient conditions for DP-4-colorability of planar graphs with cycle restrictions, using vertex identification and discharging methods.

Findings

Three new sufficient conditions for DP-4-colorability.

Results proved via vertex identification and discharging.

Conditions are expressed in a color extendability framework.

Abstract

DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization of list coloring. It was originally used to solve a longstanding conjecture by Borodin, stating that every planar graph without cycles of lengths 4 to 8 is 3-choosable. In this paper, we give three sufficient conditions for a planar graph to be DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7) are stated in the ``color extendability'' form, and uniformly proved by vertex identification and discharging method.