Pyramidal $k-$cycle decompositions

Gloria Rinaldi
University of Modena and Reggio Emilia, Italy

Marco Buratti
University of Perugia, Italy

Tommaso Traetta
Ryerson University, Toronto (ON) Canada



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. \noindent 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.

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