  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 $G$.
