Hamiltonian Cycle Systems Which Are Both Cyclic and Symmetric

Abstract

The notion of a symmetric Hamiltonian cycle system (HCS) of a graph Γ has been introduced and studied by J. Akiyama, M. Kobayashi, and G. Nakamura [J Combin Des 12 (2004), 39–45] for Γ=Kv, by R. A. Brualdi and M. W. Schroeder [J Combin Des 19 (2011), 1–15] for Γ=Kv−I, and then naturally extended by V. Chitra and A. Muthusamy [Discussiones Mathematicae Graph Theory, to appear] to the multigraphs Γ=λKv and Γ=λKv−I. In each case, there must be an involutory permutation ψ of the vertices fixing all the cycles of the HCS and at most one vertex. Furthermore, for Γ=λKv−I, this ψ 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 Kv−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 Γ 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 λK2n+1 with both properties exists if and only if 2n+1 is a prime.

A Complete Solution to the Existence of ‐Cycle Frames of Type

“…We recall, for instance, that there is no cyclic one-factorization of the complete graph of order 2 n whenever n > 2 [27], but there is a dihedral one [6]. Similarly, there is no cyclic Hamilton cycle-decomposition of K (n) 2 whenever n is divisible by 4 [30] but there is a dihedral one [11]. Proof.…”

“…Example 6.2. By following the instructions of the previous theorem, a 7-CF(4 8 ) is the one whose vertex set is those of the orbit of the holey C 7 -factor {(∞ 1 , (1, 0), (13, 1), (2, 0), (12, 1), (3, 0), (11,1)) , (∞ 4 , (1, 1), (13, 0), (2, 1), (12, 0), (3, 1), (11, 0)) , (∞ 2 , (4, 0), (10, 0), (5, 0), (6, 1), (9, 1), (8, 1)) , (∞ 3 , (4, 1), (10, 1), (5, 1), (6, 0), (9, 0), (8, 0))} of course under the rules that…”

Enumerating the Walecki-Type Hamiltonian Cycle Systems

“…. , 1) and, since Walecki-type HCSs are symmetric Hamiltonian cycle systems [3], we also have F I (x, y) = F I (x + 2n, y + 2n). Thus we can equivalently consider the invariant F(I ) = {F I (x, x + y), ∀ x, y = 1, .…”

“…Notice that and . Furthermore and, since Walecki‐type HCSs are symmetric Hamiltonian cycle systems , we also have . Thus we can equivalently consider the invariant .…”

Some Results on 1‐Rotational Hamiltonian Cycle Systems

“…In it was proved the existence of at least two nonisomorphic symmetric HCS for any (and at least three when is a prime). The necessary and sufficient conditions for the existence of a symmetric HCS(2 n ) have been determined in while the necessary and sufficient conditions for the existence of a HCS(2 n ) that is cyclic and symmetric at the same time can be found in .…”