### On r-Dynamic Coloring for Graph Operation of Cycle, Star, Complete, and Path

#### 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. By a proper k -coloring of graph G, we mean a map c : V (G) → 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))| ≥ 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 ∈ S} for a vertex subset S . The r-dynamic chromatic number, written as χ_{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 χ(G), and the 2-dynamic chromatic number of graph denoted by χ_{d} (G). By simple observation with a greedy coloring algorithm, it is easy to see that χ_{r} (G) ≤ χ_{r+1}(G), however χ_{r+1}(G) − χ_{r} (G) does not always have the same diﬀerence. Thus finding an exact values of χ_{r} (G) is significantly useful. In this paper, we investigate the some exact value of χ_{r} (G) when G is for an operation product of cycle, star, complete, and path graphs.

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