A path in an edge−colored graph is said to be a rainbow path if every edge in the path has different color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of vertices. The rainbow connection of a graph G, denoted by rc(G), is the smallest number of colors required to color the edges of graph such that the graph is rainbow connected. Given two arbitrary vertices u and v in G, a rainbow u−v geodesic in G is a rainbow u−v path of length d(u, v), where d(u, v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u−v geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by src(G), is the minimum number of colors needed to make G strongly rainbow connected. In this paper we determine the exact values of rc(G) and src(G) where G are F n and S m with n + 1 and 2m vertices, respectively.

For simple graphs G 1 and G 2 , the size Ramsey multipartite number m j (G 1 , G 2 ) is defined as the smallest natural number s such that any arbitrary two coloring of the graph K j×s using the colors red and blue, contains a red G 1 or a blue G 2 as subgraphs. In this paper, we obtain the exact values of the size Ramsey numbers m j (nK 2 , C m ) for j ≥ 2 and m ∈ {3, 4, 5, 6}.

A path in an edge colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge colored graph is rainbow connected if there exists a rainbow path between every pair of vertices. The rainbow connectivity of a graph G, denoted by rc(G) is the smallest number of colors required to edge color the graph such that the graph is rainbow connected.In this note, we determine the exact values of rc (G • H) where G or H are complete graph K n , path P n , tree T n , or wheel W n with n is an integer. Mathematics Subject Classification: 05C15, 05C40

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