Some Results on 1-Rotational Hamiltonian Cycle Systems



A Hamiltonian cycle system of math formula (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 math formula there exists a 3-perfect 1-rotational HCSmath formula. 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 math formula, there are at least math formula nonisomorphic 1-rotational (and hence symmetric) HCS(math formula).