Partitioning 3-Edge-Colored Complete Equi-Bipartite Graphs by Monochromatic Trees under a Color Degree Condition

The monochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exact value of t2(K(n1, n2, · · · , nk)). In this paper, we prove that if n ≥ 3, and K(n, n) is 3-edge-colored such that every vertex has color degree 3, then t3(K(n, n)) = 3.