Sufficient conditions on planar graphs to have a relaxed DP-$3$-colorability

TL;DR

This paper introduces a relaxed DP-coloring concept for planar graphs and proves that graphs without 4- or 6-cycles are DP-$(k,d)^*$-colorable, extending list coloring results.

Contribution

It defines a new relaxed DP-coloring framework and establishes colorability results for planar graphs lacking 4- or 6-cycles.

Findings

Planar graphs without 4- or 6-cycles are DP-$(k,d)^*$-colorable

Introduces a generalized relaxed DP-coloring concept

Extends list coloring results to DP-coloring context

Abstract

It is known that DP-coloring is a generalization of a list coloring in simple graphs and many results in list coloring can be generalized in those of DP-coloring. In this work, we introduce a relaxed DP-coloring which is a generalization if a relaxed list coloring. We also shows that every planar graph $G$ without $4$-cycles or $6$-cycles is DP-$(k,d)^*$-colorable. It follows immediately that $G$ is $(k,d)^*$-choosable.