Atomic Latin squares of order eleven

Maenhaut, B. M. and Wanless, I. M. (2004) Atomic Latin squares of order eleven. Journal of Combinatorial Designs, 12 1: 12-34. doi:10.1002/jcd.10064

Author Maenhaut, B. M.
Wanless, I. M.
Title Atomic Latin squares of order eleven
Journal name Journal of Combinatorial Designs   Check publisher's open access policy
ISSN 1063-8539
Publication date 2004-01-01
Sub-type Article (original research)
DOI 10.1002/jcd.10064
Volume 12
Issue 1
Start page 12
End page 34
Total pages 23
Editor Colbourn, C.
Place of publication United States of America
Publisher John Wiley & sons, Inc
Language eng
Subject 230101 Mathematical Logic, Set Theory, Lattices And Combinatorics
780101 Mathematical sciences
Abstract A Latin square is pan-Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i j. A Latin square is atomic if all of its conjugates are pan-Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1-factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan-Hamiltonian Latin square of order n describes a perfect 1-factorization of Kn,n, and vice versa. Perfect 1-factorizations of Kn,n can be constructed from a perfect 1-factorization of Kn+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn-square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self-orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self-orthogonal Latin squares in the same main class as a given Latin square.
Keyword Latin squares
tactical representation
self-orthogonal Latin squares
orthogonal arrays
diagonally-cyclic Latin squares
mutually orthogonal Latin squares
perfect 1-factorizations
Q-Index Code C1

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