Embedding partial Latin squares in Latin squares with many mutually orthogonal mates

Donovan, Diane; Grannell, Mike and Yazici, Emine Şule (2020). Embedding partial Latin squares in Latin squares with many mutually orthogonal mates. Discrete Mathematics, 343(6), article no. 111835.

Abstract

In this paper it is shown that any partial Latin square of order can be embedded in a Latin square of order at most which has at least mutually orthogonal mates. Further, for any , it is shown that a pair of orthogonal partial Latin squares of order can be embedded in a set of mutually orthogonal Latin squares (MOLS) of order a polynomial with respect to . A consequence of the constructions is that, if denotes the size of the largest set of MOLS of order , then . In particular, it follows that , improving the previously known lower bound .

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