r-Dynamic Chromatic Number of Graphs

An $r$-dynamic $k$-coloring of a graph $G$ is a proper vertex $k$-coloring such that the neighbors of any vertex $v$ receive at least $\min\{r,{\rm deg}(v)\}$ different colors. The $r$-dynamic chromatic number of $G$, $\chi_r(G)$, is defined as the smallest $k$ such that $G$ admits an $r$-dynamic $k$-coloring. In this paper we introduce an upper bound for $\chi_r(G)$ in terms of $r$, chromatic number, maximum degree and minimum degree. In 2001, Montgomery \cite{MR2702379} conjectured that, for a $d$-regular graph $G$, $\chi_2(G)-\chi(G)\leq 2$. In this regard, for a $d$-regular graph $G$, we present two upper bounds for $\chi_2(G)-\chi(G)$, one of them, $\lceil 5.437\log d+2.721\rceil$, is an improvement of the bound $14.06\log d +1$, proved by Alishahi (2011) \cite{MR2746973}. Also, we give an upper bound for $\chi_2(G)$ in terms of chromatic number, maximum degree and minimum degree.