Some Results on 1-Rotational Hamiltonian Cycle Systems

A Hamiltonian cycle system of the complete graph on v vertices (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 integer n greatest or equal to 3, there exists a 3-perfect 1-rotational HCS(2n+1). This allows to get the existence of an 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, an imprtant lower bound for the number of non isomorphic 1-rotational (and hence symmetric) HCSs is obtained.