Cyclic hamiltonian cycle systems of the complete graph minus a 1-factor

Merola

2012

The existence problem for a Hamiltonian cycle decomposition of K_{2n} − I (the so called cocktail party graph) with a dihedral automorphism group acting sharply transitively\ud on the vertices is completely solved. Such Hamiltonian cycle decompositions exist for all even n while, for n odd, they exist if and only if the following conditions hold: (i) n is not a prime power; (ii) there is a suitable l such that p ≡ 1 (mod 2l) for all prime factors p of n and the number of the prime factors (counted with their respective multiplicities) such that p ̸≡ 1 (mod 2l+1) is even. Thus in particular one has a dihedral Hamiltonian cycle decomposition of the cocktail party graph on 8k vertices for all k, while it is known that no such cyclic Hamiltonian cycle decomposition exists. Finally, this paper touches on a recently introduced symmetry requirement by proving that there exists a dihedral and symmetric Hamiltonian cycle system\ud of K_{2n} − I if and only if n ≡ 2 (mod 4)

8 k + 4

, it has both a dihedral and a cyclic Hamiltonian decomposition.…”

Section: Introductionmentioning

confidence: 53%

“…Indeed all its cycles have stabilizer of even order (4 or

n / 2

or n ). Thus, the following result is implicit in . Theorem If

n \equiv 2

(mod 4), then there exists a cyclic symmetric Hamiltonian cycle system of

K_{2 n} - I

.…”

Section: Concerning Symmetric Hamiltonian Cycle Systems Of K2n−imentioning

confidence: 99%

“…Note that following the construction of Lemma 3.3 in it is possible to see that

scriptS

contains every

n = p^{α} q

with

p < q

primes,

p^{α} \equiv q

(mod 4), and

α \geq 1

. In all other cases the construction of Lemma 3.3 uses several base cycles with orbit‐length 2 so that their stabilizer has odd order n and then condition (iii) of Remark is not satisfied.…”

Section: Concerning Symmetric Hamiltonian Cycle Systems Of K2n−imentioning

confidence: 99%

“…On the other hand, we have constructed some examples of a cyclic symmetric Hamiltonian cycle system of

K_{2 n} - I

even for some small values of n for which the construction in gives a nonsymmetric one. Thus, it might be the case that every admissible n (namely different from a prime power) is in

scriptS

.…”

Section: Concerning Symmetric Hamiltonian Cycle Systems Of K2n−imentioning

confidence: 99%

“…On the other hand, the existence of a cyclic Hamiltonian cycle system of

K_{2 n + 1}

and

K_{2 n} - I

is completely settled in and , respectively: there exists a cyclic Hamiltonian cycle system of

K_{2 n + 1}

if and only if the order

2 n + 1

is neither 15 nor

p^{α}

( p prime and

α > 1

), while there exists a cyclic Hamiltonian cycle system of

K_{2 n} - I

iff

n \equiv 1

or 2 (mod 4) and

n \neq p^{α}

( p an odd prime and

α \geq 1

).…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph

Merola

2012

Hamiltonian Cycle Systems Which Are Both Cyclic and Symmetric

Merola

2013

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.

Some Results on 1‐Rotational Hamiltonian Cycle Systems

Rinaldi

Traetta

2013

A Hamiltonian cycle system of the complete graph on v vertices (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 integer n greatest or equal to 3, there exists a 3-perfect 1-rotational HCS(2n+1). This allows to get the existence of an 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, an imprtant lower bound for the number of non isomorphic 1-rotational (and hence symmetric) HCSs is obtained