### On r-Dynamic Coloring of Some Special Graphs Operations

#### 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*) *!* *S*, where *jSj* = *k*, such that any two adjacent vertices receive di®erent colors. An* r*-dynamic* k*-coloring is a proper *k*-coloring *c* of *G* such that *jc*(*N*(*v*))*j ¸* *minfr; d*(*v*)*g* for each vertex *v* in *V* (*G*), where *N*(*v*) is the neighborhood of *v* and *c*(*S*) = *fc*(*v*) : *v* *2* *Sg* for a vertex subset *S*. The *r*-dynamic chromatic number, written as *Â** _{r}*(

*G*), is the minimum

*k*such that

*G*has an

*r*-dynamic

*k*-coloring. The 1-dynamic chro-matic number of graph is equal to its chromatic number, denoted by

*Â*(

*G*), and the 2-dynamic chromatic number of graph has been studied under the name a dynamic chromatic number, denoted by

*Â*

*(*

_{d}*G*). By simple observation it is easy

to see that *Â** _{r}*(

*G*)

*·*

*Â*

_{r}_{+1}(

*G*), for example

*Â*(

*C*

_{6}) = 2, but

*Â*

*(*

_{d}*C*

_{6}) = 3. In this paper we will show the exact values of some graph operation of special graphs.

#### Full Text:

PDF### Refbacks

- There are currently no refbacks.