Pseudo 2-factor isomorphic regular bipartite graphs

Abreu, M.; Diwan, Ajit A.; Jackson, Bill; Labbate, Domenico; Sheehan, J.

doi:10.1016/j.jctb.2007.08.006

Cited by 10 publications

(32 citation statements)

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“…Abreu et al [1] extended these results on 2-factor hamiltonian graphs to the more general family of pseudo 2-factor isomorphic graphs. A graph G is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of G.…”

Section: Introductionmentioning

confidence: 91%

A counterexample to the pseudo 2-factor isomorphic graph conjecture

Goedgebeur

2015

Discrete Applied Mathematics

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A graph $G$ is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of $G$. Abreu et al. conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs (Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture 3.6). Using a computer search we show that this conjecture is false by constructing a counterexample with 30 vertices. We also show that this is the only counterexample up to at least 40 vertices. A graph $G$ is 2-factor hamiltonian if all 2-factors of $G$ are hamiltonian cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite graph can be obtained from $K_{3,3}$ and the Heawood graph by applying repeated star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003, Conjecture 3.2). We verify that this conjecture holds up to at least 40 vertices.Comment: 8 pages, added some extra information in Discrete Applied Mathematics (2015

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Section: Introductionmentioning

confidence: 91%

A counterexample to the pseudo 2-factor isomorphic graph conjecture

Goedgebeur

2015

Discrete Applied Mathematics

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“…Let T 3 (n) be the graph obtained from T (n) by adding the edges u 1 1 v 3 n , u 2 1 v 1 n , u 3 1 v 2 n . In [3], Boben proved that for each fixed value of n, no two of the graphs T 1 (n), T 2 (n), T 3 (n) are isomorphic.…”

Section: Symmetric Configurations Nmentioning

confidence: 99%

“…The Pappus graph: Recall that the Levi graph of the Pappus 9 3 configuration is the following pseudo 2-factor isomorphic but not 2-factor isomorphic cubic bipartite graph [1], called the Pappus graph P 0 .…”

Section: Symmetric Configurations Nmentioning

confidence: 99%

“…[6]). We have already proved in [1,Proposition 3.3] that the Pappus graph is pseudo 2-factor isomorphic. We need to prove that all other irreducible Levi graphs are not pseudo 2-factor isomorphic and we will do so by finding in each of them a 2-factor with an odd number of circuits and another 2-factor with an even number of circuits.…”

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

confidence: 99%

“…Moreover, 2-factor isomorphic bipartite graphs are extended in [1] to the more general family of pseudo 2-factor isomorphic graphs i.e. graphs G with the property that the parity of the number of circuits in a 2-factor is the same for all 2-factors of G. Example of these graphs are K 3,3 , the Heawood graph H 0 and the Pappus graph P 0 .…”

Section: Introductionmentioning

confidence: 99%

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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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