On r-Dynamic Coloring of Neighborhood Corona of Path with Some graphs

Abstract

Consider the simple, finite, connected and undirected graph H = (V, E) in which V and E denotes the vertex set and edge set of the graph H. The r-dynamic coloring of a graph H is the proper p-coloring of the vertices of the graph H in which |c(N(a)| ≥ min{r, d(a)}, for each . The lowest p which allows H an r-dynamic coloring with p colors is called the r-dynamic chromatic number of the graph H and it is denoted as X r (H). Let H 1 and H 2 be two graphs with vertex disjoint sets of n 1 and n 2vertices. The neighborhood corona of two graphs H 1 and H 2 is obtained by taking one copy of the graph H 1 and n 1 copies of the graph H 2 and by joining each neighbor of the ith vertex of H 1 to each and every vertex of the ith copy of H 2. It is denoted as . In this paper, we determine the r-dynamic chromatic number of the neighborhood corona of path graph Pm with path Pn , complete graph Kn , cycle Cn and star graph K1,n . These graphs are denoted as and respectively.