The Analysis of r-dynamic Vertex Colouring on Graph Operation Of Shackle

Abstract

Let G be a simple, connected and undirected graph and r, k be natural numbers. An edge coloring that uses k colors is a k-edge coloring. Thus a graph G can be described as a function 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. In this paper, we will study the existence of r-dynamic k-coloring when G is shackle of wheel graph. As we know, that a shackle operation of H denoted by shack(H, v, n) is a shackle with vertex as the connector. We also can generated shackle graph with edge connector or subgraph as the connector.