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) = a2 - aq + max {a, 2q - 2}. © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 928, 2004 Part of the research of Tao Jiang was done at Michigan Technological University, Houghton, MI 49931.