Perfect one-factorizations of K22 with automorphism group of order greater than six: a complete catalogue

A complete catalogue of the perfect one-factorizations of the complete graph K22 whose automorphism group has order greater than six: 687 isomorphism classes, each given once in canonical form as its 21 factors. A one-factorization of K22 splits the 231 edges into 21 perfect matchings, called factors; it is perfect when the union of any two of them is a single Hamiltonian cycle on all 22 vertices. Two factorizations belong to the same class when some relabeling of the 22 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| > 6 and is complete for that range: every isomorphism class of perfect one-factorization of K22 whose automorphism group has order greater than six appears in the file exactly once. Classes with |Aut| from 1 to 6 are not included. The distribution by group order is 7:536, 10:126, 20:20, 21:3, 110:1, 220:1. The three one-rotational classes are #420, #582 and #625, and the seventeen classes with an automorphism of type 20 1^2 (from even starters) are listed in the README; both counts agree with Pike and Shalaby (2003). No class folds to an atomic Latin square of order 21 at any vertex. The records are sorted in ascending order of the canonical form and numbered contiguously from #1 to #687, so a record number is a position in this version of the catalogue rather than a permanent name; cite the version you used. Every record was re-verified from its matrix alone: 687 valid perfect one-factorizations, no recomputed group order disagreeing with the one recorded, and no class recorded twice. The accompanying README documents the file format, the record numbering and the checks performed. The engine, the run scripts and the argument that the searches cover every group of order greater than six are public at https://github.com/LeAndr-Graph/P1F-Census. The engine extensions for K22 and above, the run scripts and much of the documentation were developed with AI coding assistants (Google Antigravity, Anthropic Claude Code); the data were checked independently as described above.