Computational results and new bounds for the circular flow number of snarks

Goedgebeur, Jan; Mattiolo, Davide; Mazzuoccolo, Giuseppe

doi:10.1016/j.disc.2020.112026

Cited by 5 publications

(3 citation statements)

References 19 publications

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“…Máčajová and Raspaud determined all snarks with circular flow number 5 up to 30 vertices in [8]. We designed an algorithm for computing the circular flow number of a cubic graph (the details of this algorithm are described in [6]). By applying this algorithm to the complete list of all snarks up to 36 vertices from [3], we were able to determine all snarks with circular flow number 5 up to that order.…”

Section: Final Remarksmentioning

confidence: 99%

A unified approach to construct snarks with circular flow number 5

Goedgebeur

Mattiolo

Mazzuoccolo

2020

Journal of Graph Theory

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The well‐known 5‐flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5‐flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5.

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Section: Final Remarksmentioning

confidence: 99%

A unified approach to construct snarks with circular flow number 5

Goedgebeur

Mattiolo

Mazzuoccolo

2020

Journal of Graph Theory

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“…Snarks with excessive index 4 can have circular flow number 5 (see [13]), and so can be critical to Conjecture 1.1. At the same time, snarks with excessive index at least 5 can have circular flow number strictly less than 5 [19], and consequently admit a 2‐bisection by [8].…”

Section: Introductionmentioning

confidence: 99%

Ban–Linial's Conjecture and treelike snarks

Zerafa

2022

Journal of Graph Theory

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A bridgeless cubic graph G is said to have a 2-bisection if there exists a 2-vertex-colouring of G (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection.The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are known as snarks, in particular, those with excessive index at least 5, that is, whose edge set cannot be covered by four perfect matchings. Moreover, Esperet et al. state that a possible counterexample to Ban-Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2-bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this study we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2-bisection.

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“…Snarks with excessive index 4 can have circular flow number 5 (see [13]), and so can be critical to Conjecture 1.1. At the same time, snarks with excessive index at least 5 can have circular flow number strictly less than 5 [18], and consequently admit a 2-bisection by [8].…”

Section: Introductionmentioning

confidence: 99%

Ban--Linial's Conjecture and treelike snarks

Zerafa

2021

Preprint

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A bridgeless cubic graph G is said to have a 2-bisection if there exists a 2-vertexcolouring of G (not necessarily proper) such that: (i) the colour classes have the same cardinality, and (ii) the monochromatic components are either an isolated vertex or an edge. In 2016, Ban and Linial conjectured that every bridgeless cubic graph, apart from the well-known Petersen graph, admits a 2-bisection. In the same paper it was shown that every Class I bridgeless cubic graph admits such a bisection. The Class II bridgeless cubic graphs which are critical to many conjectures in graph theory are snarks, in particular, those with excessive index at least 5, that is, whose edge-set cannot be covered by four perfect matchings. Moreover, Esperet et al. state that a possible counterexample to Ban-Linial's Conjecture must have circular flow number at least 5. The same authors also state that although empirical evidence shows that several graphs obtained from the Petersen graph admit a 2-bisection, they can offer nothing in the direction of a general proof. Despite some sporadic computational results, until now, no general result about snarks having excessive index and circular flow number both at least 5 has been proven. In this work we show that treelike snarks, which are an infinite family of snarks heavily depending on the Petersen graph and with both their circular flow number and excessive index at least 5, admit a 2-bisection.

show abstract

Computational results and new bounds for the circular flow number of snarks

Cited by 5 publications

References 19 publications

A unified approach to construct snarks with circular flow number 5

A unified approach to construct snarks with circular flow number 5

Ban–Linial's Conjecture and treelike snarks

Ban--Linial's Conjecture and treelike snarks

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