Perfect one-factorizations of K18 with non-trivial automorphism group: a complete catalogue

A catalogue of the perfect one-factorizations (perfect 1-factorizations, P1Fs) of the complete graph K18 whose automorphism group is non-trivial: 10,710 isomorphism classes, each given once in canonical form as its 17 factors.

A one-factorization of K18 splits the 153 edges into 17 perfect matchings, called factors; it is perfect when the union of any two of them is a single Hamiltonian cycle on all 18 vertices. Two factorizations belong to the same class when some relabeling of the 18 vertices carries one to the other, and |Aut| is the order of the group of relabelings carrying a factorization to itself.

The catalogue covers |Aut| > 1 and is complete for that range: every isomorphism class of perfect one-factorization of K18 with a non-trivial automorphism group appears in the file exactly once. Classes with a trivial automorphism group are not included.

The distribution by group order is 2:10179, 3:351, 4:144, 8:22, 16:12, 17:1, 272:1.

Two of the classes fold to an atomic Latin square of order 17; both squares are written out in full in K18_P1F_LS17_Atomic.txt, together with the construction that produces them.

The accompanying README documents the file format, the record numbering and the checks performed on the data.