Hamilton decompositions of complete multipartite graphs with any 2-factor leave

For m ≥ 1 and p ≥ 3, given a set of integers s1, . . . , sq with sj ≥ p + 1 for 1 ≤ j ≤ q and ∑q j=1 sj = mp, necessary and sufficient conditions are found for the existence of a hamilton decomposition of the complete p-partite graph Km,...,m−E(U), where U is a 2-factor of Km,...,m consisting of q cycles, the jth cycle having length sj . This result is then used to completely solve the problem when p = 3, removing the condition that sj ≥ p + 1.