Hamiltonian cycle systems which are both cyclic and symmetric

The notion of a symmetric Hamiltonian cycle system (HCS) of a graph $\Gamma$ has been introduced and studied by J. Akiyama, M. Kobayashi, and G. Nakamura [J Combin Des 12 (2004), 39–45] for $\Gamma = K_v$, by R. A. Brualdi and M. W. Schroeder [J Combin Des 19 (2011), 1– 15] for $\Gamma = K_v − I$, and then naturally extended by V. Chitra and A. Muthusamy [Discussiones Mathematicae Graph Theory, to appear] to the multigraphs $\Gamma = \lambda K_v$ and $\Gamma = λK_v − I$. In each case, there must be an involutory permutation $\psi$ of the vertices fixing all the cycles of the HCS and at most one vertex. Furthermore, for $\Gamma = λK_v − I$, this $\psi$ should be precisely the permutation switching all pairs of endpoints of the edges of I. An HCS is cyclic if it is invariant under some cyclic permutation of all the vertices. The existence question for a cyclic HCS of $K_v − I$ has been completely solved by Jordon and Morris [Discrete Math (2008), 2440–2449]— and we note that their cyclic construction is also symmetric for v ≡ 4 (mod 8). It is then natural to study the existence problem of an HCS of a graph or multigraph $\Gamma$ as above which is both cyclic and symmetric. In this paper, we completely solve this problem: in the case of even order, the final answer is that cyclicity and symmetry can always cohabit when a cyclic solution exists. On the other hand, imposing that a cyclic HCS of odd order is also symmetric is very restrictive; we prove in fact that an HCS of $\lambda K_{2n+1}$ with both properties exists if and only if 2n + 1 is a prime.