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A family of perfect factorisations of complete bipartite graphs

A 1-factorisation of a graph is perfect if the union of any two of its
1-factors is a Hamiltonian cycle. Let n=p^{2} for an odd prime p.
We construct a family of (p-1)/2 non-isomorphic perfect
1-factorisations of K_{n,n}. Equivalently, we construct
pan-Hamiltonian Latin squares of order n. A Latin square is
pan-Hamiltonian if the permutation defined by any row relative to any
other row is a single cycle.
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