Se p 20 11 DECOMPOSITIONS OF COMPLETE MULTIPARTITE GRAPHS INTO COMPLETE GRAPHS

Let k ≥ l ≥ 1 and n ≥ 1 be integers. Let G(k, n) be the complete k-partite graph with n vertices in each colour class. An l-decomposition of G(k, n) is a set X of copies of Kk in G(k, n) such that each copy of Kl in G(k, n) is a subgraph of exactly one copy of Kk in X. This paper asks: when does G(k, n) have an l-decomposition? The answer is well known for the l = 2 case. In particular, G(k, n) has a 2-decomposition if and only if there exists k − 2 mutually orthogonal Latin squares of order n. For general l, we prove that G(k, n) has an l-decomposition if and only if there are k− l Latin cubes of dimension l and order n, with an additional property that we call mutually invertible. This property is stronger than being mutually orthogonal. An l-decomposition of G(k, n) is then constructed whenever no prime less than k divides n.