Transversals of Latin squares and covering radius of sets of permutations

Abstract

We consider the symmetric group Sn as a metric space with the Hamming metric. The covering radius cr(S) of a set of permutations S⊂Sn is the smallest r such that Sn is covered by the balls of radius r centred at the elements of S. For given n and s, let f(n,s) denote the cardinality of the smallest set S of permutations with cr(S)⩽n−s.The value of f(n,2) is the subject of a conjecture by Kézdy and Snevily that implies two famous conjectures by Ryser and Brualdi on transversals in Latin squares. We show that f(n,2)⩽n+O(logn) for all n and that f(n,2)⩽n+2 whenever n=3m for m>1. We also construct, for each odd m⩾3, a Latin square of order 3m with two rows that each contain 2m−2 transversal-free entries. This gives an infinite family of Latin squares with odd order n and at most n/3+O(1) disjoint transversals. The previous strongest upper bound for such a family was n/2+O(1).Finally, we show that f(5,3)=15 and record a proof by Blackburn that cr(AGL(1,q))=q−3 when q is odd.