Perfect $1$-factorisations of $K_{11,11}$

TL;DR

This paper reports a computer enumeration of perfect 1-factorisations of K_{11,11} and explores their connection to row-Hamiltonian Latin squares, filling gaps in existing literature.

Contribution

It provides the first enumeration of perfect 1-factorisations of K_{11,11} and links these to all row-Hamiltonian Latin squares of order 11.

Findings

Enumerated all perfect 1-factorisations of K_{11,11}

Identified all row-Hamiltonian Latin squares of order 11

Clarified the relationship between these Latin squares and classical perfect 1-factorisations

Abstract

A perfect $1$-factorisation of a graph is a decomposition of that graph into $1$-factors such that the union of any two $1$-factors is a Hamiltonian cycle. A Latin square of order $n$ is row-Hamiltonian if for every pair $(r,s)$ of distinct rows, the permutation mapping $r$ to $s$ has a single cycle of length $n$. We report the results of a computer enumeration of the perfect $1$-factorisations of the complete bipartite graph $K_{11,11}$. This also allows us to find all row-Hamiltonian Latin squares of order $11$. Finally, we plug a gap in the literature regarding how many row-Hamiltonian Latin squares are associated with the classical families of perfect $1$-factorisations of complete graphs.