# On the Hamilton-Waterloo Problem

The Oberwolfach Problem asks whether the complete graph $K_{v}$ (if $v$ is odd) or $K_{v} - I$, the complete graph with the edges of a 1-factor removed, (if $v$ is even) admits a 2-factorization in which each factor is isomorphic to a given 2-factor $F$. In the case that $F$ is uniform, i.e. all its cycles are of the same length, the Oberwolfach Problem has been settled; in particular, the case of uniform odd-cycle factors was solved in a 1989 paper by Alspach, Schellenberg, Stinson and Wagner.

In this talk, we consider the related Hamilton-Waterloo Problem, where, given two 2-factors $F_{1}$ and $F_{2}$, and nonnegative integers $\alpha$ and $\beta$, we seek to decompose $K_{v}$ or $K_{v} - I$ into $\alpha$ copies of $F_{1}$ and $\beta$ copies of $F_{2}$. We focus on the case that the factors $F_{1}$ and $F_{2}$ are uniform, consisting of $m$-cycles and $n$-cycles, respectively. While much progress has been made, certain cases remain open; the cases where $m$ and $n$ are of opposite parity or where $1 \in \{\alpha, \beta\}$ have been especially challenging. We provide an overview of the problem and discuss recent work on these cases.

This is joint work with Peter Danziger, Adrián Pastine and Tommaso Traetta.

*Bio*: Andrea Burgess completed her Ph.D. in 2009 at the University of Ottawa. She subsequently held an NSERC Postdoctoral Fellowship at Memorial University and Toronto Metropolitan University (then known as Ryerson University). She is currently an Associate Professor at the University of New Brunswick's Saint John campus. Her research interests include combinatorial design theory and graph searching. She has worked extensively on problems related to cycle decomposition, including the Oberwolfach and Hamilton-Waterloo Problems, and colourings of designs and decompositions.