# Perfect 1-Factorisations and Hamiltonian Latin Squares

A 1-*factorisation* of a graph is a decomposition of the edges of the graph into 1-factors (perfect matching). The 1-factorisation is *perfect* if the union of any two of its 1-factors is a Hamilton cycle. Kotzig famously conjectured that the complete graph $K_{2n}$ has a perfect 1-factorisation (P1F) for every positive integer $n$. Despite considerable attention, we are very far from proving this conjecture. Only 3 sparse infinite families have been constructed, together with some sporadic orders. Several exhaustive enumerations have recently revealed that there are 3155 P1Fs of $K_{16}$.

For P1Fs of the complete bipartite graph $K_{n,n}$ our state of knowledge is marginally better, with 7 sparse infinite families published. A P1F of $K_{n,n}$ is equivalent to a *row-Hamiltonian Latin square* of order $n$. These are Latin squares with no non-trivial Latin subrectangles; equivalently, the permutation which maps any row to any other row is an $n$-cycle. Each Latin square has six *conjugates* obtained by uniformly permuting its (row, column, symbol) triples. Let $\nu(L)$ denote the number of conjugates of $L$ that are row-Hamiltonian. It is easy to see that $\nu(L) \in \{0, 2, 4, 6\}$ and that $\nu = 0$ can be achieved for all $n > 3$. At the other extreme, $\nu = 6$ is achieved by the so-called *atomic* Latin squares, including the Cayley tables of cyclic groups of prime order. There is also a known infinite family with $\nu = 2$. We announce the first infinite family with $\nu = 4$. It allows us to build Latin squares in which *every* pair of rows form a Hamilton cycle and *no* pair of columns form a Hamilton cycle. As a corollary, we will answer a question on quasigroup varieties posed by Falconer in 1970.

*Bio*: Ian Wanless obtained his Ph.D. in 1998 from the Australian National University, and is currently a Professor at Monash University, in Melbourne, Australia. He has published extensively across a wide range of discrete mathematical disciplines including graph theory, design theory, enumeration and combinatorial algebra. He is a recognised authority on Latin squares and on matrix permanents. He has won the Kirkman and Hall medals of the ICA, and the Australian Mathematical Society Medal. He is one of the Editors-in-Chief of the Electronic Journal of Combinatorics.