On [formula omitted]-dynamic chromatic number of graphs

Abstract

An r-dynamic k-coloring of a graph G is a proper vertex k-coloring of G such that the neighbors of any vertex v receive at least min{r,deg(v)} different colors. The r-dynamic chromatic number of G, χr(G), is defined as the smallest k such that G admits an r-dynamic k-coloring. In this paper, first we introduce an upper bound for χr(G) in terms of r, chromatic number, maximum degree and minimum degree. Then, motivated by a conjecture of Montgomery (2001) stating that for a d-regular graph G, χ2(G)−χ(G)≤2, we prove two upper bounds for χ2−χ on regular graphs. Our first upper bound ⌈5.437logd+2.721⌉ improves a result of Alishahi (2011). Also, our second upper bound shows that Montgomery’s conjecture is implied by the existence of a χ(G)-coloring for any regular graph G, such that any two vertices whose neighbors are unicolored in this coloring, have no common neighbor.