# Generalization of some results on list coloring and DP-coloring

## TL;DR

This paper introduces DPG-coloring, a new concept combining DP-coloring and variable degeneracy, to extend and improve upon existing results in list coloring and DP-coloring of planar graphs.

## Contribution

It proposes DPG-coloring, a novel approach that generalizes previous coloring results, leading to new insights and broader applicability in graph coloring theories.

## Key findings

- Extended DP-3-coloring results for planar graphs.
- Proved DP-4-colorability under specific cycle constraints.
- Enhanced understanding of list and DP-coloring relationships.

## Abstract

In this work, we introduce DPG-coloring using the concepts of DP-coloring and variable degeneracy to modify the proofs on the following papers: (i) DP-3-coloring of planar graphs without $4$, $9$-cycles and cycles of two lengths from $\{6, 7, 8\}$ (R. Liu, S. Loeb, M. Rolek, Y. Yin, G. Yu, Graphs and Combinatorics 35(3) (2019) 695-705), (ii) Every planar graph without $i$-cycles adjacent simultaneously to $j$-cycles and $k$-cycles is DP-$4$-colorable when $\{i, j, k\}=\{3, 4, 5\}$ (P. Sittitrai, K. Nakprasit, arXiv:1801.06760(2019) preprint), (iii) Every planar graph is $5$-choosable (C. Thomassen, J. Combin. Theory Ser. B 62 (1994) 180-181). Using this modification, we obtain more results on list coloring, DP-coloring, list-forested coloring, and variable degeneracy.