A Sufficient condition for DP-4-colorability

TL;DR

This paper extends known results by establishing that planar graphs without certain cycles are 4-DP-colorable, a generalization encompassing list and signed coloring, thus broadening the understanding of graph colorability.

Contribution

It proves that planar graphs without cycles of lengths 3 to 6 are 4-DP-colorable, generalizing previous results on list and signed colorings.

Findings

Planar graphs without C_k (k=3..6) are 4-DP-colorable.

Extends known results from list and signed coloring to DP-coloring.

Provides a unified condition for DP-4-colorability in planar graphs.

Abstract

DP-coloring of a simple graph is a generalization of list coloring, and also a generalization of signed coloring of signed graphs. It is known that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-choosable. Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each $k \in \{3, 4, 5, 6\}$, every signed planar graph without $C_k$ is signed 4-choosable. In this paper, we show that for each $k \in \{3, 4, 5, 6\}$, every planar graph without $C_k$ is 4-DP-colorable, which is an extension of the above results.