Edge decomposition of complete tripartite graphs

A graph H decomposes a graph G if and only if the edges of G can be partitioned into disjoint subsets each of which induces a graph isomorphic to H. Wilson (in: C.St.J.A. Nash-Williams, J. Sheehan (Eds.), Proceedings of the Fifth British Combinatorial Conference, Aberdeen, 1975) showed, among other things, that for any graph H, there is an integer n such that H decomposes Kn, and Häggkvist (in: J. Siemons (Ed.), Surveys in Combinatories 1989, Invited papers for 12th British Combinatorial Conference, LMS Lecture Notes 141, Cambridge University Press, Cambridge, 1989, pp. 115–147) showed that for any bipartite H, there is an n such that H decomposes Kn,n. In this paper, we extend this result to tripartite graphs, by showing that for any tripartite graph H, there is an integer n such that H decomposes Kn,n,n.