Heterochromatic tree partition numbers for complete bipartite graphs

An rr-edge-coloring of a graph GG is a surjective assignment of rr colors to the edges of GG. A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number   of an rr-edge-colored graph GG, denoted by tr(G)tr(G), is the minimum positive integer p   such that whenever the edges of the graph GG are colored with rr colors, the vertices of GG can be covered by at most p   vertex-disjoint heterochromatic trees. In this paper we give an explicit formula for the heterochromatic tree partition number of an rr-edge-colored complete bipartite graph Km,nKm,n.