On list 3-dynamic coloring of near-triangulations

TL;DR

This paper investigates list 3-dynamic coloring of near-triangulation planar graphs, establishing an improved upper bound of 6 for the list 3-dynamic chromatic number, which is tighter than the general bound for all planar graphs.

Contribution

It proves that near-triangulation planar graphs have a list 3-dynamic chromatic number at most 6, improving bounds for this specific class.

Findings

ch_3^d(G)  6 for near-triangulations

Better upper bounds than general planar graphs

Enhanced understanding of dynamic coloring in specific planar graph classes

Abstract

An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring such that for any vertex $v$, there are at least $\min\{r, deg_G(v) \}$ distinct colors in $N_G(v)$. The $r$-dynamic chromatic number $\chi_r^d(G)$ of a graph $G$ is the least $k$ such that there exists an $r$-dynamic $k$-coloring of $G$. The list $r$-dynamic chromatic number of a graph $G$ is denoted by $ch_r^d(G)$. Loeb et al. $[11]$ showed that $ch_3^d(G)\leq 10$ for every planar graph $G$, and there is a planar graph $G$ with $\chi_3^d(G)= 7$. In this paper, we study a special class of planar graphs which have better upper bounds of $ch_3^d(G)$. We prove that $ch_3^d(G) \leq 6$ if $G$ is a planar graph which is near-triangulation, where a near-triangulation is a planar graph whose bounded faces are all 3-cycles.