Heterochromatic tree partition problem in complete tripartite graphs

The heterochromatic tree partition number of an r -edge-colored graph G , denoted by t r ( G ) , is the minimum positive integer p such that whenever the edges of the graph G are colored with r colors, the vertices of G 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 t r ( K n 1 , n 2 , n 3 ) of an r -edge-colored complete tripartite graph K n 1 , n 2 , n 3 . Given an r -edge-colored complete tripartite graph K n 1 , n 2 , n 3 , based on the proof of the formula, we can find at most t r ( K n 1 , n 2 , n 3 ) vertex disjoint heterochromatic trees which together cover all the vertices of K n 1 , n 2 , n 3 .