Permutation Separations and Complete Bipartite Factorisations of K_{n, n}

Suppose p < q are odd and relatively prime. In this paper we complete the proof that Kn,n has a factorisation into factors F whose components are copies of Kp,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 0 ≤ k ≤ n, there exists a sequence (π1, π2, . . . , π2k+1) of (not necessarily distinct) permutations of {1, 2, . . . , n} such that each value in {−k, 1 − k, . . . , k} occurs exactly n times as πj(i) − i for 1 ≤ j ≤ 2k − 1 and 1 ≤ i ≤ n.