Brooks-type theorem for \(r\)-hued coloring of graphs

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 \(G\).