On r-Dynamic Coloring of Some Special Graphs Operations

I.H. Agustin, da k, N.I. Wulandari

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 Âr(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 Âd(G). By simple observation it is easy

 

to see that Âr(G) · Âr +1(G), for example Â(C6) = 2, but Âd(C6) = 3. In this paper we will show the exact values of some graph operation of special graphs.

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