On r-dynamic coloring of central vertex join of path, cycle with certain graphs

Abstract

Abstract Let G = (V, E) be a simple finite connected and undirected graph with n vertices and m edges. The n vertices are assigned the colors through mapping c : V [G] → I +. An r-dynamic coloring is a proper k-coloring of a graph G such that each vertex of G receive colors in at least min{deg(υ),r} different color classes. The minimum k such that the graph G has r-dynamic k coloring is called the r-dynamic chromatic number of graph G denoted as χ r (G). Let G 1 and G 2 be a graphs with n 1 and n 2 vertices and m 1 and m 2 edges. The central vertex join of G1 and G 2 is the graph G 1   V ˙   G 2 is obtained from C(G 1) and G 2 joining each vertex of G 1 with every vertex of G 2. The aim of this paper is to obtain the lower bound for r-dynamic chromatic number of central vertex join of path with a graph G, central vertex join of cycle with a graph G and r-dynamic chromatic number of P m V ˙   P n ,   P m V ˙   K n ,   P m V ˙   K n ,   P m V ˙   C n ,   C m V ˙   K n and C m V ˙   C n respectively.