A proper coloring of the graph assigns colors to the vertices, edges, or both so that proximal elements are assigned distinct colors. Concepts and questions of graph coloring arise naturally from practical problems and have found applications in many areas, including Information Theory and most notably Theoretical Computer Science. A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color class. The b-chromatic number of a graph G, denoted by uðGÞ, is the maximal integer k such that G may have a b-coloring with k colors. In this paper, we obtain the b-chromatic number for the sun let graph S n , line graph of sun let graph LðS n Þ, middle graph of sun let graph MðS n Þ, total graph of sun let graph TðS n Þ, middle graph of wheel graph MðW n Þ and the total graph of wheel graph TðW n Þ. 1991 MATHEMATICS SUBJECT CLASSIFICATION: 05C15; 05C75; 05C76 ª 2014 Production and hosting by Elsevier B.V. on behalf of Egyptian Mathematical Society.

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) .

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.

A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by χ 2 (G). We denote the cartesian product of G and H by G H. In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph K r K s , complete graph with complete bipartite graph K n K 1,s and wheel graph with complete graph W l K n . ARTICLE HISTORY

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