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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