Determine the exact number of proper edge colorings of complete graphs

Determine the exact cardinality of Y_n, the set of proper edge colorings of the complete graph K_{2n} with 2n colors, equivalently the set of 2n pairwise disjoint matchings whose union is the edge set of K_{2n}.

Background

The paper introduces Y_n to encode maximum affine equiangular sets in X_1 through proper edge colorings of K_{2n} with 2n colors. The exact enumeration of Y_n is used to count the corresponding affine equiangular sets. Because the exact cardinality is unavailable, the authors derive the lower bound |Y_n| ≥ (2n−1)!F(n)T(2n), where F(n) counts 1-factorizations of K_{2n} and T(2n) counts matchings of K_{2n}.

References

The cardinality of $Y_n$ is the number of proper edge colorings of $K_{2n}$ with $2n$ colors. Since this is not known, we provide a lower bound as follows.

— Sets of equiangular lines in dimension $18$ constructed from $A_9 \oplus A_9 \oplus A_1$  (2503.06377 - Lin et al., 9 Mar 2025) in Section 5, Maximum affine equiangular sets in X_1