On snarks that are far from being 3-edge colorable

Hägglund, Jonas

doi:10.48550/arxiv.1203.2015

Cited by 8 publications

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“…Note that a cubic bridgeless graph has excessive index 3 if and only if it is 3-edge-colorable, and deciding the latter is NP-complete. Hägglund [8,Problem 3] asked if it is possible to give a characterization of all cubic graphs with excessive index 5. In Section 2, we prove that the structure of cubic bridgeless graphs with excessive index at least five is far from trivial.…”

Section: Introductionmentioning

confidence: 99%

“…A question raised by Fouquet and Vanherpe [6] is whether the Petersen graph is the unique snark with excessive index at least five. This question was answered by the negative by Hägglund using a computer program [8]. He proved that the smallest snark distinct from the Petersen graph having excessive index at least five is a graph H on 34 vertices (see Figure 7).…”

Section: Introductionmentioning

confidence: 99%

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

Mazzuoccolo¹

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Mazzuoccolo¹

2013

The Seventh European Conference on Combinatorics, Graph Theory and Applications

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“…In particular, this shows that any cubic bridgeless graph G whose edge-set can be covered by 4 perfect matchings satisfies m 3 (G) ≥ 5 6 . It follows that the conjecture stating that every cubic bridgeless graph G satisfies m 3 (G) ≥ 4 5 only needs to be verified for graphs whose edge-set cannot be covered by 4 perfect matchings (some results on this class of graphs can be found in [3] and [6]). Recall that Kaiser, Král', and Norine [8] proved that every cubic bridgeless graph G satisfies m 2 (G) ≥ 3 5 .…”

Section: Theorem 7 Conjecture 4 Implies That Any Cubic Bridgeless Gra...mentioning

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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“…G e can be edge-3-colored) for every edge e of G. Not all snarks are critical. (In fact some snarks G are so severely "anti-critical" that ψ(G, e) = 0 for every edge of G; for further information and earlier references on such snarks, see [BGHM,Section 4.7] and [Hä,Section 3]. ) Starting with the Petersen graph and Theorem 3.5, and applying induction using [Br2, Theorem 2.2 and subsequent sentence], one obtains Theorem 5.1 in Section 5 above with the word "snark" replaced by the phrase "critical snark".…”

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confidence: 99%

Snarks from a Kászonyi perspective: A survey

Bradley

2015

Discrete Applied Mathematics

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This is a survey or exposition of a particular collection of results and open problems involving snarks -simple "cubic" (3-valent) graphs for which, for nontrivial reasons, the edges cannot be 3-colored. The results and problems here are rooted in a series of papers by László Kászonyi that were published in the early 1970s. The problems posed in this survey paper can be tackled without too much specialized mathematical preparation, and in particular seem well suited for interested undergraduate mathematics students to pursue as independent research projects. This survey paper is intended to facilitate research on these problems.

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On snarks that are far from being 3-edge colorable

Cited by 8 publications

References 0 publications

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

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

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

Snarks from a Kászonyi perspective: A survey

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