Embedding a latin square in a pair of orthogonal latin squares

Abstract

In this paper, it is shown that a latin square of order n with n ! 3 and n 6 ¼ 6 can be embedded in a latin square of order n 2 which has an orthogonal mate. A similar result for idempotent latin squares is also presented.

A polynomial embedding of pairs of orthogonal partial Latin squares

“…In [4], Hilton et al formulate some necessary conditions for a pair of orthogonal partial latin squares to be embedded in a pair of orthogonal latin squares. Jenkins [5], considered the less difficult problem of embedding a single partial latin square in a latin square which has an orthogonal mate. His embedding was of order n 2 .…”

Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

“…Theorem 4.8. [31,32] An idempotent PLS(t), t 3, can be embedded in an idempotent LS((2t + 1) 2 ), which has an idempotent orthogonal mate.…”

“…Since not every LS has an orthogonal mate it is reasonable to return to the investigation of the embedding of a single LS in a pair of MOLS. It is this problem that Jenkins [31] addressed in 2006 proving:…”

“…Jenkins returned to PLS and used Evans' result (Theorem 3.3), to embed a PLS(t) in an LS(n), where n 2t, and subsequently applied Theorem 4.5 to prove: Theorem 4.7. [31,32] If t 4, then a PLS(t) can be embedded in an LS(4t 2 ) which has an orthogonal mate.…”

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

“…In [6], Hilton et al formulate some necessary conditions for a pair of orthogonal partial Latin squares to be embedded in a pair of orthogonal Latin squares. Then in [7] Jenkins developed a construction for embedding a single partial Latin square of order n in a Latin square of order 4n 2 for which there exists an orthogonal mate. In 2014, Donovan and Yazıcı developed a construction that verified that a pair of orthogonal partial Latin squares, of order n, can be embedded in a pair of orthogonal Latin squares of order at most 16n 4 .…”