Dynamic Chromatic Number of Bipartite Graphs

Abstract

<p>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 χ<sub>2</sub>(G). Montgomery conjectured that for every r-regular graph G, χ<sub>2</sub>(G)-χ(G) ≤ 2 . Finding an optimal upper bound for χ<sub>2</sub>(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 χ<sub>2</sub>(G)-χ(G) ≤ 2⌈(Δ(G))/(δ(G))⌉. It was shown that χ<sub>2</sub>(G)-χ(G) ≤ α’ (G) +k* [2]. Also, χ<sub>2</sub>(G)-χ(G) ≤ α(G) +k* [1]. We prove that if G is a simple graph with δ(G)>2, then χ<sub>2</sub>(G)-χ(G) ≤ (α’ (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.</p>