# Cyclic Hamiltonian cycle systems for the complete multipartite graph

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Francesca Merola

Roma Tre University, Italy

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Anita Pasotti

Universit\`a degli Studi di Brescia, Italy

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Marco A. Pellegrini

Universit\`a Cattolica del Sacro Cuore, Italy

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Michael W. Schroeder

Marshall University, USA

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**Minisymposium:**
GENERAL SESSION TALKS

**Content:**
A Hamiltonian cycle system (HCS) for a graph or multigraph $\Gamma$ is a set $\cal B$ of Hamiltonian cycles of $\Gamma$
whose edges partition the edge set of $\Gamma$. A cycle system is {\em regular} if there is an automorphism group $G$ of the graph $\Gamma$ acting sharply transitively on the vertices of $\Gamma$ and permuting the cycles of ${\cal B}$, and it is called {\em cyclic} if $G$ is the cyclic group.
The existence problem for cyclic HCS for the complete graph $K_n$, $n$ odd, and for the graph $K_{2n}-I$, $I$ a 1-factor, (the so-called {\em cocktail party graph}), has been solved by Buratti and Del Fra (2004), and Jordon and Morris (2008) respectively.
In the talk I will consider existence results for cyclic Hamiltonian cycle systems for $K_{m\times n}$, the complete multipartite graph with $m$ parts, each of size $n$.
I will present necessary and sufficient conditions for the existence of a cyclic HCS for the graph $K_{m\times n}$ when the number of parts $m$ is even, and more generally for $mn$ even, and discuss some work in progress for the case in which both $m$ and $n$ are odd.
I will also touch on the symmetric HCS introduced by Brualdi and Schroeder in 2011 for the cocktail party graph, and recently generalized by Schroeder to complete multipartite graphs;
indeed we may note that cyclic cycle systems often turn out to possess this additional symmetry requirement,
so that it makes sense to discuss these results in this context.