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

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 a ∈ V ( H ) . 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 2 vertices. 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 i th vertex of H 1 to each and every vertex of the i th copy of H 2 . It is denoted as H 1 ⋄ H 2 . In this paper, we determine the r -dynamic chromatic number of the neighborhood corona of path graph P m with path P n , complete graph K n , cycle C n and star graph K 1,n . These graphs are denoted as P m ⋄ P n , P m ⋄ K n , P m ⋄ C n and P m ⋄ K 1 , n respectively.