# On the Hamilton-Waterloo problem for a class of Cayley graphs

### Tommaso Traetta Ryerson University, Toronto (ON) Canada

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Minisymposium: GENERAL SESSION TALKS

Content: The Hamilton--Waterloo problem HWP($\Gamma; m,n; \alpha$, $\beta$) asks for a factorization of the simple graph $\Gamma$ into $\alpha$ $C_m$--factors and $\beta$ $C_n$--factors. The classic version of the problem, that is, the case in which $\Gamma$ is the complete graph is still open, although it has been the subject of an extensive research activity. In this talk, I will consider the Cayley graph $C_m[n]=Cay(\mathbb{Z}_{m}\times \mathbb{Z}_{n}, S)$ with connection set $S=\{1,-1\}\times \mathbb{Z}_{n}$ and present an almost complete solution to HWP($C_m[n]; m,n; \alpha,\beta$) with $m$ and $n$ odd. Andrea Burgess will show how this result can be used to make progress on the classic problem. This is joint work with Andrea Burgess and Peter Danziger.

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