On rr-dynamic chromatic number of graphs

An rr-dynamic kk-coloring of a graph GG is a proper vertex kk-coloring of GG such that the neighbors of any vertex vv receive at least min{r,deg(v)}min{r,deg(v)} different colors. The rr-dynamic chromatic number of GG, χr(G)χr(G), is defined as the smallest kk such that GG admits an rr-dynamic kk-coloring. In this paper, first we introduce an upper bound for χr(G)χr(G) in terms of rr, chromatic number, maximum degree and minimum degree. Then, motivated by a conjecture of Montgomery (2001) stating that for a dd-regular graph GG, χ2(G)−χ(G)≤2χ2(G)−χ(G)≤2, we prove two upper bounds for χ2−χχ2−χ on regular graphs. Our first upper bound ⌈5.437logd+2.721⌉⌈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)χ(G)-coloring for any regular graph GG, such that any two vertices whose neighbors are unicolored in this coloring, have no common neighbor.