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

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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.