Near-Automorphisms of Latin Squares

作者：Cavenagh Nicholas J^*; Stones Douglas S

来源：Journal of Combinatorial Designs, 2011, 19(5): 365-377.

DOI：10.1002/jcd.20282

• intercalate
• Latin square
• partial orthomorphism
• near-automorphism

We define a near-automorphism alpha of a Latin square L to be an isomorphism such that L and alpha L differ only within a 2 x 2 subsquare. We prove that for all n >= 2 except n is an element of {3,4}, there exists a Latin square which exhibits a near-automorphism. We also show that if alpha has the cycle structure (2, n-2), then L exists if and only if n equivalent to 2 (mod 4), and can be constructed from a special type of partial orthomorphism. Along the way, we generalize a theorem by Marshall Hall, which states that any Latin rectangle can be extended to a Latin square. We also show that if alpha has at least 2 fixed points, then L must contain two disjoint non-trivial subsquares.

• 出版日期2011-9