Abstract

• An edge-colored graph G is called a rainbow graph if all the colors on its edges are distinct. Let  be a family graph of an edge-colored graph G such as   G. The rainbow graph denoted by rb G( , )  is related to the anti-Ramsey number AR G( , )  . The anti-Ramsey AR G( , ),  introduced by Erdős et al., is the maximum number of colors in an edge coloring graph of G without rainbow copy of any graph in G. Evidently, rb G AR G ( , ) ( , ) 1     , rb G( , )  is the rainbow number of  in any edge coloring graph G. In this paper, we consider the existence of rainbow number with independent cycles in the complete bipartite graphs, denoted by Km,n, order m and n with bipartitions (M, N). For this result, we endeavor to construct the complete bipartite graphs on the multi-graphs without independent cycles. Denote that the rainbow number rb (Km,n, ) for m n   5 . Let 2 denote the family of graphs containing two independent cycles. The rainbow number rb (Km,n, ) is the minimum number of colors such that  2 K m n, , then any edge coloring of Km,n with at least distinct c colors contains a rainbow copy of 2  . Without loss of generality, we obtain the result for any m n   5, rb (Km,n, ) = 3m + n – 2. Finally, we hope the main result will be supported at the fiber optic communications network in real life for our country.