On the difference between chromatic number and dynamic chromatic number of graphs

A proper vertex k-coloring of a graph G is called dynamic, if there is no vertex nu is an element of V(G) with d(nu) >= 2 and all of its neighbors have the same color. The smallest integer k such that G has a k-dynamic coloring is called the dynamic chromatic number of G and denoted by chi(2)(G). We say that nu is an element of V(G) in a proper vertex coloring of G is a bad vertex if d(nu) >= 2 and only one color appears in the neighbors of nu. In this paper, we show that if G is a graph with the chromatic number at least 6, then there exists a proper vertex chi(G)-coloring of G such that the set of bad vertices of G is an independent set. Also, we provide some upper bounds for chi(2)(G) - chi(G) in terms of some parameters of the graph G. (C) 2011 Elsevier B.V. All rights reserved.