Path factorizations of complete multipartite graphs (Q1296982)

A \(P_k\)-factor of a graph \(G\) is a spanning subgraph such that each component of it is a path on \(k\) vertices. A graph has a \(P_k\)-factorization if its edges can be partitioned into \(P_k\)-factors. It is shown that the necessary conditions \(mn\equiv 0\pmod k\) and \((m-1)nk\equiv 0\pmod{2(k-1)}\) for the wreath product of \(K_m\) and \(\overline K_n\) to have a \(P_k\)-factorization are sufficient when \(k= p+1\), \(p\) a prime, where \(\overline K_n\) is the complement of the complete graph \(K_n\).