Bipartite anti-Ramsey numbers of cycles

We determine the maximum number of colors in a coloring of the edges of Km;n such that every cycle of length 2k contains at least two edges of the same color. One of our main tools is a result on generalized path covers in balanced bipartite graphs. For positive integers q a, let g(a;q) be the maximum number of edges in a spanning subgraph G of Ka;a such that the minimum number of vertex-disjoint even paths and pairs of vertices from distinct partite sets needed to cover V (G) is q. We prove that g(a,q)1⁄4 a aq þmax {a; 2q 2} . 2004 Wiley Periodicals, Inc. J Graph Theory 47: 9–28,