Several classes of graphs and their r-dynamic chromatic numbers

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) → 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))| ≥ 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 χr(G), is the minimum k such that G has an r-dynamic k-coloring. By simple observation it is easy to see that χr(G) ≤ χr+1(G), however χr+1(G) − χr(G) does not always show a small difference for any r. Thus, finding an exact value of χr(G) is significantly useful. In this paper, we will study some of them especially when G are prism graph, three-cyclical ladder graph, joint graph and circulant graph.