On Parsimonious Edge-Colouring of Graphs with Maximum Degree Three

Fouquet, Jean-Luc; Vanherpe, Jean-Marie

doi:10.1007/s00373-012-1145-3

Cited by 11 publications

(31 citation statements)

References 12 publications

(31 reference statements)

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“…The following result from Fouquet and Vanherpe is a structural property about non 3edge-colorable cubic graphs [11]. It will be used several times in the proofs of Section e 1 e 2 e 3 Figure 2: A 2-factor containing three edges e 1 , e 2 and e 3 being in different odd cycles which induce a subgraph containing a path of length 5 (dashed lines: edges in the path of length 5, thick lines: edges from the 2-factor).…”

Section: Sets Of Type I and Iimentioning

confidence: 99%

“…Figure 2 illustrates three edges in the forbidden configuration described in Proposition 2.1 in a fixed 2-factor of a cubic graph (by Proposition 2.1, it means that there exists a 2-factor containing at most two odd cycles in the graph from Figure 2). Note that the previous proposition was not explicitly presented in the paper of Fouquet and Vanherpe but can be easily obtained by combining Properties 3 and 6 of [11,Theorem 4]. Note also that the proof of Theorem 4 is written in another paper [10] from the same authors.…”

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confidence: 99%

“…Note that the previous proposition was not explicitly presented in the paper of Fouquet and Vanherpe but can be easily obtained by combining Properties 3 and 6 of [11,Theorem 4]. Note also that the proof of Theorem 4 is written in another paper [10] from the same authors.…”

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confidence: 99%

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On S-packing edge-colorings of cubic graphs

Gastineau

Togni

2019

Discrete Applied Mathematics

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Given a non-decreasing sequence S = (s 1 , s 2 , . . . , s k ) of positive integers, an Spacking edge-coloring of a graph G is a partition of the edge set of G into k subsets {X 1 , X 2 , . . . , X k } such that for each 1 ≤ i ≤ k, the distance between two distinct edges e, e ′ ∈ X i is at least s i + 1. This paper studies S-packing edgecolorings of cubic graphs. Among other results, we prove that cubic graphs having a 2-factor are (1, 1, 1, 3, 3)-packing edge-colorable, (1, 1, 1, 4, 4, 4, 4, 4)-packing edgecolorable and (1, 1, 2, 2, 2, 2, 2)-packing edge-colorable. We determine sharper results for cubic graphs of bounded oddness and 3-edge-colorable cubic graphs and we propose many open problems.

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Section: Sets Of Type I and Iimentioning

confidence: 99%

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confidence: 99%

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On S-packing edge-colorings of cubic graphs

Gastineau

Togni

2019

Discrete Applied Mathematics

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“…From Proposition 2.2(iv), the first author [30] also managed to prove the following theorem (the most difficult case was N = 16 and it was rediscovered later by Fouquet [37]): Some of the aforementioned results were also considered for subcubic graphs by Fouquet and Vanherpe [40], and Rizzi [96].…”

Section: Basic Results and Multipolesmentioning

confidence: 99%

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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“…By the way, we do not know example of cyclically 5-edge connected snarks (excepted the Petersen graph) with a 2-factor of induced cycles of length 5. We have proposed in [3] the following problem.…”

Section: Fulkerson Coverings More Examplesmentioning

confidence: 99%

On Fulkerson conjecture

Fouquet

Vanherpe

2011

Discuss. Math. Graph Theory

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If G is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a Fulkerson covering) with the property that every edge of G is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A FR-triple is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples.Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.1991 Mathematics Subject Classification. 035 C.

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On Parsimonious Edge-Colouring of Graphs with Maximum Degree Three

Cited by 11 publications

References 12 publications

On S-packing edge-colorings of cubic graphs

On S-packing edge-colorings of cubic graphs

Measures of Edge-Uncolorability of Cubic Graphs

On Fulkerson conjecture

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