One‐factorizations of complete graphs with vertex‐regular automorphism groups

Bonisoli, Arrigo; Labbate, Domenico

doi:10.1002/jcd.1025

Cited by 27 publications

(42 citation statements)

References 14 publications

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“…A factorisation (K n , E) is said to be (G, M)-homogeneous if M < G ≤ S n , M is transitive on V and fixes each factor setwise, while G leaves E invariant and permutes the factors transitively. Since elements of G induce isomorphisms between the factors, all factors are isomorphic, and indeed 'isomorphic factorisations' of complete graphs have been well studied, see for example [3,4,13,14]. Homogeneous factorisations of complete graphs were introduced in [21] (and for graphs in general in [12]).…”

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confidence: 99%

Homogeneous factorisations of complete graphs with edge-transitive factors

Lim

Praeger

2008

J Algebr Comb

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A factorisation of a complete graph K n is a partition of its edges with each part corresponding to a spanning subgraph (not necessarily connected), called a factor. A factorisation is called homogeneous if there are subgroups M < G ≤ S n such that M is vertex-transitive and fixes each factor setwise, and G permutes the factors transitively. We classify the homogeneous factorisations of K n for which there are such subgroups G, M with M transitive on the edges of a factor as well as the vertices. We give infinitely many new examples.

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confidence: 99%

Homogeneous factorisations of complete graphs with edge-transitive factors

Lim

Praeger

2008

J Algebr Comb

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“…It is also affirmative if G is abelian, and it is not cyclic of order 2n ¼ 2 t > 4, (Buratti [4]), and if G is dihedral (Bonisoli and Labbate [1]). The answer is affirmative for each arbitrary finite non abelian 2-group admitting a cyclic subgroup of index 2 (Bonisoli and Rinaldi [2]).…”

Section: Introductionmentioning

confidence: 97%

“…G must be the semi-direct product of Z 2 with its normal complement, and G always realizes a 1-factorization of K 2n upon which it acts sharply transitively on vertices, see [1,Remark 1].…”

Section: Introductionmentioning

confidence: 99%

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Nilpotent 1-factorizations of the complete graph

Rinaldi

2005

J. Combin. Designs

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For which groups G of even order 2n does a 1-factorization of the complete graph on 2n veritces exist with the property of admitting G as a sharply vertex-transitive automorphism group? The complete answer is still unknown. Using the definition of a starter in G introduced in [M. Buratti "Abelian 1-factorizations of the complete graph" Europ. J Comb. 2001, pp.291-295], we give a positive answer for new classes of groups; for example, the nilpotent groups with either an abelian Sylow 2-subgroup or a non-abelian Sylow 2-subgroup which possesses a cyclic subgroup of index 2. Further considerations are given in case the automorphism group G fixes a 1-factor

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Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph

Buratti

Merola

2012

J of Combinatorial Designs

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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)

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One‐factorizations of complete graphs with vertex‐regular automorphism groups

Cited by 27 publications

References 14 publications

Homogeneous factorisations of complete graphs with edge-transitive factors

Homogeneous factorisations of complete graphs with edge-transitive factors

Nilpotent 1-factorizations of the complete graph

Dihedral Hamiltonian Cycle Systems of the Cocktail Party Graph

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