Near‐automorphisms of Latin squares
NJ Cavenagh, DS Stones - Journal of Combinatorial Designs, 2011 - Wiley Online Library
Journal of Combinatorial Designs, 2011•Wiley Online Library
We define a near‐automorphism α of a Latin square L to be an isomorphism such that L and
αL differ only within a 2× 2 subsquare. We prove that for all n≥ 2 except n∈{3, 4}, there
exists a Latin square which exhibits a near‐automorphism. We also show that if α has the
cycle structure (2, n− 2), then L exists if and only if n≡ 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 …
αL differ only within a 2× 2 subsquare. We prove that for all n≥ 2 except n∈{3, 4}, there
exists a Latin square which exhibits a near‐automorphism. We also show that if α has the
cycle structure (2, n− 2), then L exists if and only if n≡ 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 …
Abstract
We define a near‐automorphism α of a Latin square L to be an isomorphism such that L and αL differ only within a 2 × 2 subsquare. We prove that for all n≥2 except n∈{3, 4}, there exists a Latin square which exhibits a near‐automorphism. We also show that if α has the cycle structure (2, n − 2), then L exists if and only if n≡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 α has at least 2 fixed points, then L must contain two disjoint non‐trivial subsquares. Copyright © 2011 John Wiley & Sons, Ltd. 19:365‐377, 2011
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