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 $\min{r,{\rm deg}(v)}$differentcolors\mathrm{.}The$r$-dynamicchromaticnumberof$G$,$\chi_r(G)$,isdefinedasthesmallest$k$suchthat$G$admitsan$r$-dynamic$k$-coloring\mathrm{.}Inthispaperweintroduceanupperboundfor$\chi_r(G)$ in terms of $r$, chromatic number, maximum degree and minimum degree. In 2001, Montgomery \cite{MR2702379} conjectured that, for a $d$-regular graph $G$, ${\chi }_2(G)-\chi (G)\le 2$. In this regard, for a $d$-regular graph $G$, we present two upper bounds for ${\chi }_2(G)-\chi (G)$, one of them, $\lceil 5.437\log d+2.721\rceil$,isanimprovementofthebound$14.06\log d +1$, 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.