An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min r, d(v) , for each v ∈ V(G), where N(v) and d(v) denote the neighborhood and the degree of v, respectively. The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the -dynamic chromatic number of middle, total, and central of helm graph, where = min v∈V (G) d (v) .

Let [Formula: see text] be a simple, connected undirected graph with [Formula: see text] vertices and [Formula: see text] edges. Let vertex coloring [Formula: see text] of a graph [Formula: see text] be a mapping [Formula: see text], where [Formula: see text] and it is [Formula: see text]-colorable. Vertex coloring is proper if none of the any two neighboring vertices receives the similar color. An [Formula: see text]-dynamic coloring is a proper coloring such that [Formula: see text] min[Formula: see text], for each [Formula: see text][Formula: see text]. The [Formula: see text]-dynamic chromatic number of a graph [Formula: see text] is the minutest coloring [Formula: see text] of [Formula: see text] which is [Formula: see text]-dynamic k-colorable and denoted by [Formula: see text]. By a simple view, we exhibit that [Formula: see text], howbeit [Formula: see text] cannot be arbitrarily small. Thus, finding the result of [Formula: see text] is useful. This study gave the result of [Formula: see text]-dynamic chromatic number for the central graph, Line graph, Subdivision graph, Line of subdivision graph, Splitting graph and Mycielski graph of the Flower graph [Formula: see text] denoted by [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] respectively.

An r-dynamic coloring of a graph G is a proper coloring of G such that every vertex in V(G) has neighbors in at least $\min\{d(v),r\}$ different color classes. The r-dynamic chromatic number of graph G denoted as $\chi_r (G)$, is the least k such that G has a coloring. In this paper we obtain the r-dynamic chromatic number of the central graph, middle graph, total graph, line graph, para-line graph and sub-division graph of the comb graph $P_n\odot K_1$ denoted by $C(P_n\odot K_1), M(P_n\odot K_1), T(P_n\odot K_1), L(P_n\odot K_1), P(P_n\odot K_1)$ and $S(P_n\odot K_1)$ respectively by finding the upper bound and lower bound for the r-dynamic chromatic number of the Comb graph.

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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 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 H 1 ⋄ H 2 . 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 P m ⋄ P n , P m ⋄ K n , P m ⋄ C n and P m ⋄ K 1 , n respectively.