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For positive integers m,n, the greatest number of colors that can appear in an edge coloring of Km,n which avoids rainbow cycles is m+n−1. Here these colorings are constructively characterized; it turns out that these colorings can be encoded by certain vertex labelings of full binary trees with m+ n leafs.

Edge Colorings of K(m,n) with m+n-1 Colors Which Forbid Rainbow Cycles