r-Dynamic Chromatic Number of Graphs

Abstract

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 $\min \{r,\mathrm{deg}(v)\}$ 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 $G$, ${\chi }_2(G)-\chi (G)\le 2$. In this regard, for a $d$-regular graph $G$, we present two upper bounds for ${\chi }_2(G)-\chi (G)$, one of them, $\lceil 5.437\log d+2.721\rceil$, 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.

arXiv comment9 pages

References in corpus (1)

• Upper bounds for the 2-hued chromatic number of graphs in terms of the independence number