On $r$-Hued Coloring of Corona Product of Some Graphs

Abstract

An $r$-hued coloring of a graph $G$ is a proper $k$-coloring of graph $G$ such that the neighbours of any vertex $u$ accept at least $\min \{r, d(u)\}$ different colors. The $r$-hued chromatic number is the minimum $k$ such that graph $G$ has an $r$-hued $k$-coloring, it is denoted by $\chi_r(G)$. The corona product $G\circ H$ of two graphs $G$ and $H$ is obtained by taking one copy of $G$ and $|V\left(G)\right|$ copies of $H$ and joining every vertex of $i^{th}$ copy of $H$ to the $i^{th}$ copy of $G$, where $1\le i\leq |V\left(G\right)|$. In this paper, we obtain the $r$-hued chromatic number of corona product of two graphs $G$ and $H$, denoted by $G\circ H$. First, we consider $G\circ H$, where $G$ is the path graph and $H$ is any simple graph like complete graph. Secondly, we consider $G$ as the complete graph and $H$ as the path graph. Finally we consider $G$ as the cycle graph and $H$ as the path graph.