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Keywords:

  • 1-rotational Hamiltonian cycle system;
  • binary group;
  • terrace;
  • directed terrace;
  • symmetric terrace;
  • i-perfect cycle decomposition

Abstract

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