Snarks and Flow-Critical Graphs

Silva, Cândida Nunes da; Pesci, Lissa; Lucchesi, Cláudio L.

doi:10.1016/j.endm.2013.10.047

Cited by 8 publications

(8 citation statements)

References 10 publications

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“…The hard part of the problem, however, is the snarks that cannot be obtained by a series of I-extensions from a smaller snark in such a way that each member of the extension series is a snark. Such snarks indeed exist and have already been studied [11,12,38,43]; in fact, they have been rediscovered several times [14,16,42]: they are known as critical snarks and are characterised by the property that for each edge, the inverse of I-extension produces a colourable graph.…”

Section: Introductionmentioning

confidence: 99%

Morphology of Small Snarks

Mazák

Rajník

Škoviera

2022

Electron. J. Combin.

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This paper classifies all snarks up to order $36$ and explains the reasons for their uncolourability. The crucial part of our approach is a computer-assisted structural analysis of cyclically $5$-connected critical snarks, which is justified by the fact that every other snark can be constructed from them by a series of simple operations while preserving uncolourability. Our results reveal that most of the analysed snarks are built from pieces of the Petersen graph and can be naturally distributed into a small number of classes having the same reason for uncolourability. This sheds new light on the structure of all small snarks. Based on our analysis, we generalise certain individual snarks to infinite families and identify a rich family of cyclically $5$-connected critical snarks.

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Section: Introductionmentioning

confidence: 99%

Morphology of Small Snarks

Mazák

Rajník

Škoviera

2022

Electron. J. Combin.

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“…Nevertheless, our main result shows that for snarks flow-criticality does not bring anything substantially new. There have been a number of papers following either of the two approaches to the criticality; see for example [5,13,19,20] and [6,9,10], respectively. In several other works, critical snarks have emerged in forms different from those explained above, yet in all the cases the definitions turn out to be equivalent to one of those given above.…”

Section: Introductionmentioning

confidence: 99%

“…As suggested by several authors, the idea of nontriviality of snarks is rather subtle and seems to be best captured by various reductions and decompositions of snarks [4,5,16,18]. The concept of a snark reduction is, in turn, closely related to that of criticality of a snark, which naturally takes one of two forms: criticality with respect to the non-existence of a 3-edge-colouring [3,5,16] and criticality with respect to the non-existence of a nowherezero 4-flow [6,10,8,9]. The purpose of the present paper is to show that these two types of criticality coincide.…”

Section: Introductionmentioning

confidence: 99%

Critical and flow-critical snarks coincide

Máčajová¹,

Škoviera²

2021

Discuss. Math. Graph Theory

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Over the past twenty years, critical and bicritical snarks have been appearing in the literature in various forms and in different contexts. Two main variants of criticality of snarks have been studied: criticality with respect to the non-existence of a 3-edge-colouring and criticality with respect to the non-existence of a nowherezero 4-flow. In this paper we show that these two kinds of criticality coincide, thereby completing previous partial results of de Freitas et al. [Electron. Notes Discrete Math. 50 (2015), 199-204] and Fiol et al. [ arXiv:1702.07156v1 (2017].

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“…The hard part of the problem, however, are the snarks that cannot be obtained by a series of I-extensions from a smaller snark in such a way that each member of the extension series is a snark. Such snarks indeed exist and have been already studied [11,12,37,42], in fact, they have been rediscovered several times [14,16,41]: they are known as critical snarks and are characterised by the property that for each edge the inverse of I-extension produces a colourable graph.…”

Section: Introductionmentioning

confidence: 99%

Morphology of small snarks

Mazák¹,

Rajník²,

Škoviera³

2021

Preprint

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The aim of this paper is to classify all snarks up to order 36 and explain the reasons of their uncolourability. The crucial part of our approach is a computerassisted structural analysis of cyclically 5-connected critical snarks, which is justified by the fact that every other snark can be constructed from them by a series of simple operations while preserving uncolourability. Our results reveal that most of the analysed snarks are built up from pieces of the Petersen graph and can be naturally distributed into a small number of classes having the same reason for uncolourability. This sheds new light on the structure of all small snarks. Based on our analysis, we generalise certain individual snarks to infinite families and identify a rich family of cyclically 5-connected critical snarks.

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Snarks and Flow-Critical Graphs

Cited by 8 publications

References 10 publications

Morphology of Small Snarks

Morphology of Small Snarks

Critical and flow-critical snarks coincide

Morphology of small snarks

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