## On $r$-dynamic Coloring of Operation Product of Cycle and Path Graphs

##### Abstract

Let $G$ be a simple, connected and undirected
graph. Let $r,k$ be natural numbers. By a proper $k$-coloring of a
graph $G$, we mean a map $ c : V (G) \rightarrow S$, where $|S| =
k$, such that any two adjacent vertices receive different colors. 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. It was introduced by Montgomery. Note
that 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 has been studied under the name a dynamic
chromatic number, denoted by $\chi_d(G)$. By simple observation 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 will show some exact values of $\chi_r(G)$
when $G$ is an operation product of cycle and path graphs.

##### Collections

- MIPA [81]

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