On 2-pyramidal Hamiltonian cycle systems

A Hamiltonian cycle system of the complete graph minus a 1–factor $K_2v − I$ (briefly, an HCS(2v)) is 2-pyramidal if it admits an automorphism group of order 2v − 2 fixing two vertices. In spite of the fact that the very first example of an HCS(2v) is very old and 2-pyramidal, a thorough investigation of this class of HCSs is lacking. We give first evidence that there is a strong relationship between 2-pyramidal HCS(2v) and 1-rotational Hamiltonian cycle systems of the complete graph $K_2v−1$. Then, as main result, we determine the full automorphism group of every 2-pyramidal HCS(2v). This allows us to obtain an exponential lower bound on the number of non-isomorphic 2-pyramidal HCS(2v).