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 v ∈ V ( G ) with d ( v ) ≥ 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 χ 2 ( G ) . We say that v ∈ V ( G ) in a proper vertex coloring of G is a bad vertex if d ( v ) ≥ 2 and only one color appears in the neighbors of v . In this paper, we show that if G is a graph with the chromatic number at least 6 , then there exists a proper vertex χ ( G ) -coloring of G such that the set of bad vertices of G is an independent set. Also, we provide some upper bounds for χ 2 ( G ) − χ ( G ) in terms of some parameters of the graph G .