Heterochromatic tree partition number of a complete multipartite graph

A heterochromatic tree is an edge-colored tree in which any two edges have different colors. The heterochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum k such that whenever the edges of the graph G are colored with r colors, the vertices of G can be covered by at most k vertexdisjoint heterochromatic trees. In this paper we determine the heterochromatic tree partition number of an r-edge-colored complete multipartite graph Kn1,n2,··· ,nk . Though we cannot give explicit expression for the heterochromatic tree partition number of Kn1,n2,··· ,nk , we describe two types of canonical r-edge-colorings φ ∗ r,1 and φ∗r,2 of Kn1,n2,··· ,nk , from which we can obtain the number in polynomial time. As corollaries, for complete graphs and complete bipartite graphs we give explicit expression for the heterochromatic tree partition number.