Variable degeneracy of graphs with restricted structures

TL;DR

This paper introduces structural results for specific classes of graphs, proving their weak degeneracy and existence of certain transversals, which leads to improved bounds on their DP-paint number and list vertex arboricity.

Contribution

It establishes new structural properties and reducibility results for three classes of graphs, enhancing bounds on their coloring and degeneracy parameters.

Findings

All three graph classes are weakly 3-degenerate.

They have a strictly f-degenerate transversal.

DP-paint number at most four, list vertex arboricity at most two.

Abstract

Bernshteyn and Lee defined a new notion, weak degeneracy, which is slightly weaker than the ordinary degeneracy. It is proved that strictly $f$-degenerate transversal is a common generalization of list coloring, $L$-forested-coloring and DP-coloring. In this paper, we consider three classes of graphs, including planar graphs without any configuration in Fig. 2, toroidal graphs without any configuration in Fig. 5, and planar graphs without intersecting $5$-cycles. We give structural results for each class of graphs, and prove each structure is reducible for weakly $3$-degenerate and the existence of strictly $f$-degenerate transversals. As consequences, these three classes of graphs are weakly $3$-degenerate, and have a strictly $f$-degenerate transversal. Then these three classes of graph have DP-paint number at most four, and have list vertex arboricity at most two. This greatly improve all the results in [2-4, 11-13, 16-18, 22, 25, 32, 34]. Furthermore, the first and the third classes of graphs have Alon-Tarsi number at most four.