Dynamic Chromatic Number of Regular Graphs

TL;DR

This paper investigates the difference between the chromatic number and the dynamic chromatic number of regular graphs, proving bounds and confirming conjectures for graphs with diameter at most 2, and providing counterexamples to previous conjectures.

Contribution

It establishes new bounds on the dynamic chromatic number of regular graphs, confirms conjectures for graphs with diameter at most 2, and disproves a prior conjecture with counterexamples.

Findings

Confirmed the conjecture for regular graphs with diameter at most 2 and chromatic number at least 4.

Proved that the difference between dynamic and chromatic number is at most 6ln(k)+2 for k-regular graphs.

Constructed regular graphs with chromatic number n where the difference is at least 1.

Abstract

A dynamic coloring of a graph $G$ is a proper coloring such that for every vertex $v\in V(G)$ of degree at least 2, the neighbors of $v$ receive at least 2 colors. It was conjectured [B. Montgomery. {\em Dynamic coloring of graphs}. PhD thesis, West Virginia University, 2001.] that if $G$ is a $k$-regular graph, then $\chi_2(G)-\chi(G)\leq 2$. In this paper, we prove that if $G$ is a $k$-regular graph with $\chi(G)\geq 4$, then $\chi_2(G)\leq \chi(G)+\alpha(G^2)$. It confirms the conjecture for all regular graph $G$ with diameter at most 2 and $\chi(G)\geq 4$. In fact, it shows that $\chi_2(G)-\chi(G)\leq 1$ provided that $G$ has diameter at most 2 and $\chi(G)\geq 4$. Moreover, we show that for any $k$-regular graph $G$, $\chi_2(G)-\chi(G)\leq 6\ln k+2$. Also, we show that for any $n$ there exists a regular graph $G$ whose chromatic number is $n$ and $\chi_2(G)-\chi(G)\geq 1$. This result gives a negative answer to a conjecture of [A. Ahadi, S. Akbari, A. Dehghan, and M. Ghanbari. \newblock On the difference between chromatic number and dynamic chromatic number of graphs. \newblock {\em Discrete Math.}, In press].