Abstract
A subgraph H of an edge-colored graph G is rainbow if all of its edges have different colors. The anti-Ramsey number is the maximum number of colors in an edge-coloring of G with no rainbow copy of H. Originally a complete graph was considered as G. In this paper, we consider a complete bipartite graph as the host graph and discuss some results for the graph H being hamiltonian cycle and perfect matching. Let $$c(K_{p,p},t)$$ and $$m(K_{p,p},t)$$ be the maximum number of colors in an edge-coloring of the complete bipartite graph $$K_{p,p}$$ not having t edge-disjoint rainbow hamiltonian cycles and perfect matchings, respectively. We prove that $$c(K_{p,p},t)=p^2-p+t$$ for $$t \ge 2$$ , $$p\ge 4t-1$$ and $$m(K_{p,p},t)=p^2-2p+t+1$$ for $$t \ge $$ 2, $$p\ge 2t+8$$ .
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