For positive integers m, n, the greatest number of colors that can appear in an edge coloring of K(m,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.
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Johnson, Peter and Zhang, Claire
"Edge Colorings of K(m,n) with m+n-1 Colors Which Forbid Rainbow Cycles,"
Theory and Applications of Graphs: Vol. 4
, Article 1.
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol4/iss1/1