2-Factor hamiltonian graphs

Funk, Martin; Jackson, Bill; Labbate, Domenico; Sheehan, J.

doi:10.1016/s0095-8956(02)00031-x

Cited by 28 publications

(53 citation statements)

References 10 publications

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“…Note 2: It is conjectured in [2] that a k-regular bipartite graph G is 2-factor hamiltonian if and only if either k ¼ 2 and G is a circuit, or k ¼ 3 and G can be obtained from K 3;3 and H 0 by repeated star products.…”

Section: Preliminariesmentioning

confidence: 99%

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Regular bipartite graphs with all 2-factors isomorphic

Aldred

Funk

Jackson

et al. 2004

Journal of Combinatorial Theory, Series B

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Section: Preliminariesmentioning

confidence: 99%

“…We call such graphs 2-factor hamiltonian. In [2] the following result was proved for 2-factor hamiltonian graphs. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

Regular bipartite graphs with all 2-factors isomorphic

Aldred

Funk

Jackson

et al. 2004

Journal of Combinatorial Theory, Series B

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“…[6]). We have already proved in [1,Proposition 3.3] that the Pappus graph is pseudo 2-factor isomorphic.…”

Section: Theorem 3 the Heawood And The Pappus Graphs Are The Only Irrmentioning

confidence: 99%

“…Several recent papers have addressed the problem of characterizing families of graphs (particularly regular graphs) which have these properties. It is shown in [2,6] that k-regular 2-factor isomorphic bipartite graphs exist only when k ∈ {2, 3} and an infinite family of 3-regular 2-factor hamiltonian bipartite graphs, based on K 3,3 and the Heawood graph, is constructed in [6]. It is conjectured in [6] that every 3-regular 2-factor hamiltonian bipartite graph belongs to this family.…”

Section: Introductionmentioning

confidence: 99%

Irreducible pseudo 2-factor isomorphic cubic bipartite graphs

Abreu

Labbate²,

Sheehan³

2011

Des. Codes Cryptogr.

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A bipartite graph is pseudo 2-factor isomorphic if the number of circuits in each 2-factor of the graph is always even or always odd. We proved (Abreu et al., J Comb Theory B 98:432–442, 2008) that the only essentially\ud 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graph of girth 4 is$ K_{3,3}$, and conjectured (Abreu\ud et al., 2008, Conjecture 3.6) that the only essentially 4-edge-connected cubic bipartite graphs are $K_{3,3}$, the Heawood graph and the Pappus graph. There exists a characterization of symmetric configurations $n _3$ due to Martinetti (1886) in which all symmetric configurations $n_3$ can be obtained from an infinite set of so called irreducible configurations (Martinetti, Annali di Matematica Pura ed Applicata II 15:1–26, 1888). The list of irreducible configurations has been completed by Boben (Discret Math 307:331–344, 2007) in terms of their irreducible Levi graphs. In this paper we characterize irreducible pseudo 2-factor isomorphic cubic bipartite\ud graphs proving that the only pseudo 2-factor isomorphic irreducible Levi graphs are the Heawood and Pappus graphs. Moreover, the obtained characterization allows us to partially prove the above Conjecture

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“…In this paper, we consider extending k independent edges to 2-factors. Our immediate interest [1] is to develop a reduction technique in order to study graphs, which have unique hamiltonian 2-factors. We mean by this that such a graph is hamiltonian and all its 2-factors are connected.…”

Section: Motivationmentioning

confidence: 99%