Heterochromatic tree partition problem in complete tripartite graphs

The heterochromatic tree partition number   of an rr-edge-colored graph GG, denoted by tr(G)tr(G), is the minimum positive integer pp such that whenever the edges of the graph GG are colored with rr colors, the vertices of GG can be covered by at most pp vertex disjoint heterochromatic trees. In this paper, we give an explicit formula for the heterochromatic tree partition number tr(Kn1,n2,n3)tr(Kn1,n2,n3) of an rr-edge-colored complete tripartite graph Kn1,n2,n3Kn1,n2,n3. Given an rr-edge-colored complete tripartite graph Kn1,n2,n3Kn1,n2,n3, based on the proof of the formula, we can find at most tr(Kn1,n2,n3)tr(Kn1,n2,n3) vertex disjoint heterochromatic trees which together cover all the vertices of Kn1,n2,n3Kn1,n2,n3.