On 3-Hued Coloring of Graphs

Abstract:

      For integers $k>0$, $r>0$, a $(k,r)$-coloring of a graph $G$ is a proper $k$-coloring of the vertices such that every vertex of degree $d$ is adjacent to vertices with at least $\min\{d,r\}$ different colors. The $r$-hued chromatic number, denoted by $\chi_r(G)$, is the smallest integer $k$ for which a graph $G$ has a $(k,r)$-coloring. Define a graph $G$ is $r$-normal, if $\chi_r(G)=\chi(G)$. In this paper, we present two sufficient conditions for a graph to be $3$-normal, and the best upper bound of $3$-hued chromatic number of a certain families of graphs.