On $r$-Dynamic Coloring for Operation Product of Cycle and Cycle Graphs

Abstract

For integer $k,r>0,(k,r)$ -coloring of graph $G$ is a proper coloring on the vertices of $G$ by $k$-colors such that every vertex $v$ of degree $d(v)$ is adjacent to vertices with at least $min\{d(v),r\}$ different color. Graph coloring provides a model. By a proper $k$ -coloring of graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $|S| = k$, such that any two adjacent vertices are different color. An $r$ -dynamic $k$ -coloring is a proper $k$ -coloring $c$ of $G$ such that $|c(N (v))| \geq min\{r, d(v)\}$ for each vertex $v$ in $V(G)$, where $N(v)$ is the neighborhood of $v$ and $c(S) = \{c(v) : v \in S\}$ for a vertex subset $S$ . The $r$-dynamic chromatic number, written as $\chi_r(G)$, is the minimum $k$ such that $G$ has an $r$-dynamic $k$-coloring. Note the $1$-dynamic chromatic number of graph is equal to its chromatic number, denoted by $\chi(G)$, and the $2$-dynamic chromatic number of graph denoted by $\chi_d(G)$. By simple observation with a greedy coloring algorithm, it is easy to see that $\chi_r(G)\le \chi_{r+1}(G)$, however $\chi_{r+1}(G)-\chi_r(G)$ does not always have the same difference. Thus finding an exact values of $\chi_r(G)$ is significantly useful. In this paper, we investigate the some exact value of $\chi_r(G)$ when $G$ is for an operation product of cycle and cycle graphs.