Dynamic Chromatic Number of Bipartite Graphs

A dynamic coloring of a graph G is a proper vertex coloring such that for every vertex v ∈ V (G) of degree at least 2, the neighbors of v receive at least 2 colors. The smallest integer k such that G has a dynamic coloring with k colors, is called the dynamic chromatic number of G and denoted by χ2(G). Montgomery conjectured that for every r-regular graph G, χ2(G) − χ(G) ≤ 2 [19]. Finding an optimal upper bound for χ2(G) − χ(G) seems to be an intriguing problem. We show that there is a constant d such that every bipartite graph G with δ(G) ≥ d, has χ2(G) − χ(G) ≤ 2⌈ ∆(G) δ(G) ⌉. It was shown that χ2(G) − χ(G) ≤ α (G) + k [2]. Also, χ2(G) − χ(G) ≤ α(G) + k ∗ [1]. We prove that if G is a simple graph with δ(G) > 2, then χ2(G) − χ(G) ≤ α ′(G)+ω(G) 2 +k . Among other results, we prove that for a given bipartite graph G = [X,Y ], determining whether G has a dynamic 4coloring l : V (G) → {a, b, c, d} such that a, b are used for part X and c, d are used for part Y is NP-complete.