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

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 G₁ and G₂ be a graphs with n₁ and n₂ vertices and m₁ and m₂ edges. The central vertex join of G₁ and G₂ is the graph ${G_1}\,\dot V\,{G_2}$ is obtained from C(G₁) and G₂ joining each vertex of G₁ with every vertex of G₂. 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 ${P_m}\dot V\,{P_n},\,{P_m}\dot V\,{K_n},\,{P_m}\dot V\,{K_n},\,{P_m}\dot V\,{C_n},\,{C_m}\dot V\,{K_n}$ and ${C_m}\dot V\,{C_n}$ respectively.