A Hamiltonian cycle system of K_v (briefly, a HCS(v)) is 1-rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any n>2 there exists a 3-perfect 1-rotational HCS. This allows to get the existence of another infinite class of 3-perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1-rotational HCS under G is G itself unless the HCS is the 2-transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion a formula by Bailey, Ollis, and Preece on the number of inequivalent 1-rotational HCSs under G. This leads us to a formula counting all 1-rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, it allows us to say that, for any , there are at least nonisomorphic 1-rotational (and hence symmetric) HCS(2n+1).

### Some results on 1-rotational Hamiltonian cycle systems

#### Abstract

A Hamiltonian cycle system of K_v (briefly, a HCS(v)) is 1-rotational under a (necessarily binary) group G if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any n>2 there exists a 3-perfect 1-rotational HCS. This allows to get the existence of another infinite class of 3-perfect (but not Hamiltonian) cycle decompositions of the complete graph. Then we prove that the full automorphism group of a 1-rotational HCS under G is G itself unless the HCS is the 2-transitive one. This allows us to give a partial answer to the problem of determining which abstract groups are the full automorphism group of a HCS. Finally, we revisit and simplify by means of a careful group theoretic discussion a formula by Bailey, Ollis, and Preece on the number of inequivalent 1-rotational HCSs under G. This leads us to a formula counting all 1-rotational HCSs up to isomorphism. Though this formula heavily depends on some parameters that are hard to compute, it allows us to say that, for any , there are at least nonisomorphic 1-rotational (and hence symmetric) HCS(2n+1).
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2014
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11391/1223203`
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