# Perfect 1-factorisations of $K_{16}$

## TL;DR

This paper presents a comprehensive computer enumeration of perfect 1-factorisations of the complete graph K_{16}, revealing new automorphism groups, invariants, and implications for Latin squares and larger graph orders.

## Contribution

It provides the first complete enumeration of P1Fs of K_{16}, introduces a new invariant for distinguishing them, and explores their properties related to Latin squares and larger orders.

## Key findings

- Found 3155 P1Fs of K_{16}
- 89 P1Fs have non-trivial automorphism groups
- No atomic Latin squares of order 15 from these P1Fs

## Abstract

We report the results of a computer enumeration that found that there are 3155 perfect 1-factorisations (P1Fs) of the complete graph $K_{16}$. Of these, 89 have a non-trivial automorphism group (correcting an earlier claim of 88 by Meszka and Rosa). We also (i) describe a new invariant which distinguishes between the P1Fs of $K_{16}$, (ii) observe that the new P1Fs produce no atomic Latin squares of order 15 and (iii) record P1Fs for a number of large orders that exceed prime powers by one.