On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings

Mazzuoccolo, Giuseppe

doi:10.1007/978-88-7642-475-5_8

Cited by 17 publications

(37 citation statements)

References 18 publications

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“…It is NP-complete to decide for a cubic bridgeless graph G whether χ e (G) = 3, since this property is equivalent to 3-edge-colourability. Similarly, Esperet and Mazzuoccolo [7] proved that it is NP-complete to decide whether χ e (G) ≤ 4, as well as to decide whether χ e (G) = 4.…”

Section: Introductionmentioning

confidence: 93%

Treelike Snarks

Abreu¹,

Kaiser²,

Labbate³

et al. 2016

Electron. J. Combin.

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We study snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hägglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we prove that the snarks from this family (we call them treelike snarks) have circular flow number φ C (G) ≥ 5 and admit a 5-cycle double cover.

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Section: Introductionmentioning

confidence: 93%

Treelike Snarks

Abreu¹,

Kaiser²,

Labbate³

et al. 2016

Electron. J. Combin.

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“…Without loss of generality, suppose that E(P ) ∩ M 1 = ∅ and both end-vertices of P are incident with an edge from E (1,2) . Thus P is a component of H 1 .…”

Section: Combining This Inequality Withmentioning

confidence: 99%

Petersen Cores and the Oddness of Cubic Graphs

Jin

Steffen

2016

Journal of Graph Theory

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“…A cubic bridgeless graph G satisfies m 3 (G) = 1 if and only if it is 3-edgecolorable, and deciding this is a well-known NP-complete problem (see [7]). It was proved in [3] that deciding whether a cubic bridgeless graph G satisfies m 4 (G) = 1 is also an NP-complete problem.…”

Section: Complexitymentioning

confidence: 99%

On the maximum fraction of edges covered by t perfect matchings in a cubic bridgeless graph

Mazzuoccolo

2015

Discrete Mathematics

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A conjecture of Berge and Fulkerson (1971) states that every cubic bridgeless graph contains 6 perfect matchings covering each edge precisely twice, which easily implies that every cubic bridgeless graph has three perfect matchings with empty intersection (this weaker statement was conjectured by Fan and Raspaud in 1994). Let m t be the supremum of all reals α ≤ 1 such that for every cubic bridgeless graph G, there exist t perfect matchings of G covering a fraction of at least α of the edges of G. It is known that the Berge-Fulkerson conjecture is equivalent to the statement that m 5 = 1, and implies that m 4 = 14 15 and m 3 = 4 5 . In the first part of this paper, we show that m 4 = 14 15 implies m 3 = 4 5 , and m 3 = 4 5 implies the Fan-Raspaud conjecture, strengthening a recent result of Tang, Zhang, and Zhu. In the second part of the paper, we prove that for any 2 ≤ t ≤ 4 and for any real τ lying in some appropriate interval, deciding whether a fraction of more than (resp. at least) τ of the edges of a given cubic bridgeless graph can be covered by t perfect matching is an NP-complete problem. This resolves a conjecture of Tang, Zhang, and Zhu.

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On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings

Abstract: International audienc

Cited by 17 publications

References 18 publications

Treelike Snarks

Treelike Snarks

Petersen Cores and the Oddness of Cubic Graphs

On the maximum fraction of edges covered by t perfect matchings in a cubic bridgeless graph

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