A perfect one-factorization of a finite simple graph is a one-factorization such that
the union of any two distinct factors is a Hamiltonian
cycle (Gill and Wanless 2020, p. 177). For a positive
integer 
,
a one-factorization of the complete
graph 
consists of 
perfect matchings. An isomorphism
between two such factorizations is a permutation
of the vertices that carries the collection of factors
of one to the collection of factors of the other.
Ivanov and Tertitski (2026, Theorem 1) report exactly 10,710 perfect one-factorizations, counted up to isomorphism, of 
having a nontrivial automorphism
group. Of these, 531 have automorphism group
of group order greater than 2 and 10,179 have automorphism
group of group order 2. This is a count restricted
to factorizations with nontrivial automorphisms,
not a count of all perfect one-factorizations of 
. The authors used AI assistants to write and optimize
parts of the enumeration software and checked the results. The exhaustive enumeration
has not been independently repeated here.
."
Bull. Austral. Math. Soc. 101, 177-185, 2020. 
with a Non-Trivial Automorphism Group." 1 Nov 2026.