Brooks-type theorem for $r$-hued coloring of graphs

Abstract

An $r$-hued coloring of a simple graph $G$ is a proper coloring of its vertices such that every vertex $v$ is adjacent to at least $\min\{r, \deg(v)\}$ differently colored vertices. The minimum number of colors needed for an $r$-hued coloring of a graph $G$, the $r$-hued chromatic number, is denoted by $\chi_{r}(G)$. In this note we show that $$\chi_r(G) \leq (r - 1)(\Delta(G) + 1) + 2,$$ for every simple graph $G$ and every $r \geq 2$, which in the case when $r < \Delta(G)$ improves the presently known $\Delta(G)$-based upper bound on $\chi_r(G)$, namely $r \Delta(G) + 1$. We also discuss the existence of graphs whose $r$-hued chromatic number is close to $(r-1)(\Delta + 1 ) + 2$ and we prove that there is a bipartite graph of maximum degree $\Delta$ whose $r$-hued chromatic number is $(r-1)\Delta + 1$ for every $r \in \{2, \dots, 9\}$ and infinitely many values of $\Delta \geq r + 2$; we believe that $(r-1)\Delta(G) + 1$ is the best upper bound on the $r$-hued chromatic number of any bipartite graph $G$.