Abstract:
Let $G$ be a simple, connected and undirected
graph. Let $r,k$ be natural number. 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. 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
$\chi_r(G)\le \chi_{r+1}(G)$, for example $\chi(C_6)=2 but \chi_d(C_6)=3$.
In this paper we will show the exact values of some graph
operation of special graphs.
