On the dynamic coloring of graphs

A dynamic coloring of a graph GG is a proper coloring such that, for every vertex v∈V(G)v∈V(G) of degree at least 2, the neighbors of vv receive at least 2 colors. In this paper, we present some upper bounds for the dynamic chromatic number of graphs. In this regard, we shall show that, for every kk-regular graph GG, χ2(G)≤χ(G)+14.06lnk+1χ2(G)≤χ(G)+14.06lnk+1. Also, we introduce an upper bound for the dynamic list chromatic number of regular graphs.