Treelike Snarks

Abreu, M.; Kaiser, Tomáš; Labbate, Domenico; Mazzuoccolo, Giuseppe

doi:10.37236/6008

Cited by 23 publications

(41 citation statements)

References 23 publications

(47 reference statements)

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“…The following theorem shows that r 2 (G) is also a measure of edge-uncolorability. (ii) 1 2 r(G) ≤ r 2 (G) ≤ min{ 2 3 r(G), 1 2 ω(G)}, and the bounds are attained.…”

Section: Max 2-and 3-colorable Subgraphsmentioning

confidence: 97%

“…The chains P (1,3) (x 2 , y 2 ) and P (1,2) (x 3 , y 3 ) are defined analogously. By Lemma 2.4 of [104] the three paths P (2,3) (x 1 , y 1 ), P (1,3) (x 2 , y 2 ) and P (1,2) Interchange the colors on the chains; i.e., consider P (3,1) (x 1 , y 3 ) and P (3,2) (x 2 , x 3 ), to obtain a proper 4-edge-coloring c of G. Then e 3 can be colored with color 3, and we still have a proper coloring. Hence r(G) < 3, a contradiction.…”

Section: -Factorsmentioning

confidence: 98%

“…In other words, a possible counterexample for the 5-cycle double cover conjecture should be look for in the class of cubic graphs with excessive index 5. Abreu, Kaiser, Labbate, and Mazzuoccolo [1] suggest a relation between excessive index and circular flow number of a cubic graph (see Section 4 for a definition). In particular, it is remarked that all known snarks with excessive index 5 have circular flow number at least 5.…”

Section: -Factorsmentioning

confidence: 99%

“…[105]. There are infinitely many snarks with circular flow number 5 (see Máčajova and Raspaud [81], and Esperet, Mazzuoccolo, and Mkrtchyan [28]), and also with further properties (see Abreu, Kaiser, Labbate, and Mazzuoccolo [1]). From the results of Kochol [72,76] and Mazzuoccolo and Steffen [88] it follows that a minimal counterexample to Conjecture 1.1 is cyclically 6-edge connected snark with girth at least 11 and with oddness at least 6.…”

Section: Nowhere-zero Flowsmentioning

confidence: 99%

See 3 more Smart Citations

Measures of Edge-Uncolorability of Cubic Graphs

Fiol

Mazzuoccolo

Steffen

2018

Electron. J. Combin.

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There are many hard conjectures in graph theory, like Tutte's 5-flow conjecture, and the 5-cycle double cover conjecture, which would be true in general if they would be true for cubic graphs. Since most of them are trivially true for 3-edge-colorable cubic graphs, cubic graphs which are not 3-edge-colorable, often called snarks, play a key role in this context. Here, we survey parameters measuring how far apart a non 3-edge-colorable graph is from being 3-edgecolorable. We study their interrelation and prove some new results. Besides getting new insight into the structure of snarks, we show that such measures give partial results with respect to these important conjectures. The paper closes with a list of open problems and conjectures.

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“…The following theorem shows that r 2 (G) is also a measure of edge-uncolorability. (ii) 1 2 r(G) ≤ r 2 (G) ≤ min{ 2 3 r(G), 1 2 ω(G)}, and the bounds are attained.…”

Section: Max 2-and 3-colorable Subgraphsmentioning

confidence: 97%

Section: -Factorsmentioning

confidence: 98%

Section: -Factorsmentioning

confidence: 99%

Section: Nowhere-zero Flowsmentioning

confidence: 99%

See 2 more Smart Citations

Measures of Edge-Uncolorability of Cubic Graphs

Fiol

Mazzuoccolo

Steffen

2018

Electron. J. Combin.

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“…The interested reader might find more information about snarks in, for example, [24] or [35]. Snarks with circular flow number 5 are known (see [3,17,20,31]) and no snark without a 2-bisection, other than the Petersen graph, has been found to date. Conjecture 1.1 is strongly related to Ando's Conjecture (ie, Conjecture 1.3) and the following holds: Proposition 2.6.…”

Section: Ban-linial's Conjecturementioning

confidence: 99%

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

Abreu

Goedgebeur

Labbate

et al. 2019

Journal of Graph Theory

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A k‐bisection of a bridgeless cubic graph G is a 2‐colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes ( monochromatic components in what follows) have order at most k. Ban and Linial Conjectured that every bridgeless cubic graph admits a 2‐bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph G with ∣ E ( G ) ∣ ≡ 0 ( mod 2 ) has a 2 ‐edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (ie, a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we provide evidence of a strong relation of the conjectures of Ban‐Linial and Wormald with Ando's Conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above‐mentioned conjectures. Moreover, we prove Ban‐Linial's Conjecture for cubic‐cycle permutation graphs. As a by‐product of studying 2‐edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.

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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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Treelike Snarks

Cited by 23 publications

References 23 publications

Measures of Edge-Uncolorability of Cubic Graphs

Measures of Edge-Uncolorability of Cubic Graphs

Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

A unified approach to construct snarks with circular flow number 5

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