Atomic latin squares of order eleven
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-factorisation 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-factorisation of Kn,n, and vice
versa. Perfect 1-factorisations of Kn,n can be constructed
from a perfect 1-factorisation 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 MOLS
involving that square. In the process 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 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 forms of a Latin square.
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