| Popis: |
A Hamiltonian cycle system of Kv (briefly, a HCS(v)) is 1-rotational under a (nec- essarily binary )g roupG if it admits G as an automorphism group acting sharply transitively on all but one vertex. We first prove that for any n ≥ 3 there exists a 3-perfect 1-rotational HCS(2n + 1). 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 auto- morphism 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 n ≥ 6, there are at least 2 � 3n/4� nonisomorphic |
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