Dynamic chromatic number of regular graphs

Abstract

A k-dynamic coloring of a graph G is a proper coloring of G with k colors such that for every vertex v∈V(G) of degree at least 2, the neighbors of v receive at least 2 colors. The dynamic chromatic number of a graph G, χ2(G), is the least number k such that G admits a k-dynamic coloring. It was conjectured [B. Montgomery, Dynamic coloring of graphs, Ph.D. Thesis, West Virginia University, 2001.] that if G is a k-regular graph, then χ2(G)−χ(G)≤2. In this paper, we prove that if G is a k-regular graph with χ(G)≥4, then χ2(G)≤χ(G)+α(G2). It confirms the conjecture for all regular graphs with diameter at most 2 and chromatic number at least 4. In fact, it shows that χ2(G)−χ(G)≤1 provided that G has diameter at most 2 and χ(G)≥4. Moreover, we show that for any k-regular graph G with no induced C4, χ2(G)−χ(G)≤2⌈4lnk+1⌉. Also, we show that for any n there exists a regular graph G whose chromatic number is n and χ2(G)−χ(G)≥1. This result gives a negative answer to a conjecture of Ahadi et al. [A. Ahadi, S. Akbari, A. Dehghan, M. Ghanbari, On the difference between chromatic number and dynamic chromatic number of graphs, Discrete Math. Available online 2011].