Dynamic Chromatic Number of Bipartite Graphs - S. Saqaeeyan; E. Mollaahamdi (Scientific Annals of Computer Science)
A dynamic coloring of a graph G is a proper vertex coloring such that for every vertex v Î 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 c2(G). Montgomery conjectured that for every r-regular graph G, c2(G)-c(G) ≤ 2 . Finding an optimal upper bound for c2(G)-c(G) seems to be an intriguing problem. We show that there is a constant d such that every bipartite graph G with d(G) ³ d , has c2(G)-c(G) ≤ 2é(D(G))/(d(G))ù. It was shown that c2(G)-c(G) ≤ a' (G) +k* . Also, c2(G)-c(G) ≤ a(G) +k* . We prove that if G is a simple graph with d(G)>2, then c2(G)-c(G) ≤ (a' (G)+w(G) )/2 +k* . Among other results, we prove that for a given bipartite graph G=[X,Y], determining whether G has a dynamic 4-coloring 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.