# Pyramidal $k-$cycle decompositions

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Gloria Rinaldi

University of Modena and Reggio Emilia, Italy

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Marco Buratti

University of Perugia, Italy

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Tommaso Traetta

Ryerson University, Toronto (ON) Canada

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**Minisymposium:**
GENERAL SESSION TALKS

**Content:**
A $k-$cycle system of order $v$ is a decomposition of the complete graph on $v$ vertices into
cycles of length $k$. An automorphism group of the system is a permutation group on the vertex
set preserving the decomposition into cycles. If we ask the group to satisfy
some particular requests, then we have two problems, strongly connected.
We ask which groups are admissible and we ask which is the spectrum of admissible values
$v$ and $k$ for which a $k$-cycle system of order $v$ with such an automorphism group exists.
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We focus our attention on the particular situation of an automorphism group
$G$ whose action is $t-$pyramidal on the vertex set, i.e., $G$ fixes $t$ vertices and has a
sharply transitive action on the remaining ones. The cases $t=0$ and $t=1$ are well known
and largely studied in literature, especially when the group is cyclic or the length of the
cycles is $3$, even though many open questions still remain.
\noindent
Here we examine the case $t > 1$ in some details. We point out some structural general properties
of the group and we completely determine the spectrum of values $v$ for which a $3-$pyramidal
$3-$cycle decomposition of order $v$ exists.