On the dynamic coloring of graphs

Let G be a graph. A proper vertex coloring of G is said to be a dynamic coloring if for every v ∈ V (G) of degree at least 2, the neighbors of v receive at least two different colors. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number of G and is denoted by χ2(G). It was conjectured that if G is an r-regular graph, then χ2(G)−χ(G) ≤ 2. In this talk, we show that the conjecture is true for all bipartite r-regular graphs. Also, we prove that if G 6∈ {C4, C5, Kk,k} is a srtongly regular graph, then χ2(G) − χ(G) ≤ 1. Among the other results, it is shown that if G is a graph with ∆(G) ≥ 3, then ch2(G) ≤ ∆(G) + 1, where ch2(G) is the list dynamic chromatic number. This result is a generalization of a theorem due to Lai, Montgomery and Poon which says that if G is a graph with ∆(G) ≥ 3, then χ2(G) ≤ ∆(G) + 1.