Permutation Separations and Complete Bipartite Factorisations of K_n,n - Martin, Nigel; Stong, Richard (The Electronic Journal of Combinatorics, 2003)
Suppose p < q are odd and relatively prime. In this paper we complete the proof that K_n,n has a factorisation into factors F whose components are copies of K_p,q if and only if n is a multiple of pq(p+q). The final step is to solve the "c-value problem" of Martin. This is accomplished by proving the following fact and some variants: For any 0kn, there exists a sequence (π_1,π_2, …,π_2k+1) of (not necessarily distinct) permutations of {1,2,…,n} such that each value in {-k,1-k,…,k} occurs exactly n times as π_j(i)-i for 1j2k-1 and 1in.