The smallest nontrivial snarks of oddness 4

Goedgebeur, Jan; Máčajová, Edita; Škoviera, Martin

doi:10.1016/j.dam.2019.09.020

Cited by 5 publications

(5 citation statements)

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“…In fact, we prove the following stronger and more detailed result which will also be needed in our next paper [23].…”

Section: Decomposition Theoremsmentioning

confidence: 68%

“…The most symmetric of them is shown in Figure 4. We will describe and analyse these 31 snarks in the sequel of this paper [23], where we also prove that they constitute a complete list of all snarks with oddness at least 4, cyclic connectivity 4, and minimum number of vertices. As we have already mentioned in Section 1 Introduction, Theorem 1.1 does not yet determine the smallest order of a nontrivial snark with oddness 4, because there might exist snarks with oddness at least 4 of order 38, 40, or 42 with cyclic connectivity greater than 4.…”

Section: Remarks and Open Problemsmentioning

confidence: 99%

“…The most symmetric of them is shown in Figure 4. We will describe and analyse these 31 snarks in the sequel of this paper [23], where we also prove that they constitute a complete list of all snarks with oddness at least 4, cyclic connectivity 4, and minimum number of vertices. are not known, however the well-known conjecture of Jaeger and Swart [31] that there are no cyclically 7-edge-connected snarks would imply that these values are not defined.…”

Section: Remarks and Open Problemsmentioning

confidence: 99%

“…We will continue our investigation of the smallest snarks with oddness at least 4 and cyclic connectivity 4 in the sequel of this paper [23]. We will display a set of 31 such snarks, analyse their properties, and prove that they constitute the complete set of snarks with oddness at least 4 and cyclic connectivity 4 on 44 vertices.…”

Section: Introductionmentioning

confidence: 98%

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Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

2019

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The family of snarks -connected bridgeless cubic graphs that cannot be 3-edge-coloured -is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468-488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.

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“…In fact, we prove the following stronger and more detailed result which will also be needed in our next paper [23].…”

Section: Decomposition Theoremsmentioning

confidence: 68%

Section: Remarks and Open Problemsmentioning

confidence: 99%

“…The most symmetric of them is shown in Figure 4. We will describe and analyse these 31 snarks in the sequel of this paper [23], where we also prove that they constitute a complete list of all snarks with oddness at least 4, cyclic connectivity 4, and minimum number of vertices. are not known, however the well-known conjecture of Jaeger and Swart [31] that there are no cyclically 7-edge-connected snarks would imply that these values are not defined.…”

Section: Remarks and Open Problemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 98%

See 2 more Smart Citations

Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

2019

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“…Using this result they proved a lower bound on the number of removable edges in a cyclically 4-connected cubic graph [1]. Later, Goedgebeur et al constructed and classified all snarks with cyclic connectivity 4 and oddness 4 up to order 44 [2,3].…”

Section: Introducitonmentioning

confidence: 99%