Petersen-colorings and some families of snarks

Hägglund, Jonas; Steffen, Eckhard

doi:10.26493/1855-3974.288.11a

Cited by 15 publications

(11 citation statements)

References 8 publications

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“…i is a cycle of length 5 of S-degree 1, 6 written v ′ i u 1 u 2 u 3 u 4 such that u 2 is its unique vertex incident to an edge in S. If C u ′ 2 is a cycle of S-degree 2 then C sends 1/5 to C u ′ 2 through v i .…”

Section: (R1)mentioning

confidence: 99%

“…Despite original approaches [10][11][12], few progress has been made on the Petersen colouring conjecture: ways to infirm it remain elusive as possible counter-examples must be snarks, that is bridgeless cubic graphs that are not 3-edge-colourable (the ones we know are usually obtained from well-structured graph operations, for which the Petersen colouring conjecture can be verified [2,6]), and confirming the conjecture is expected to be a difficult task since as reported earlier this would confirm several difficult and most researched graph conjectures.…”

mentioning

confidence: 99%

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Variations on the Petersen Colouring Conjecture

Pirot¹,

Sereni

Škrekovski³

2020

Electron. J. Combin.

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A. The Petersen colouring conjecture states that every bridgeless cubic graph admits an edge-colouring with 5 colours such that for every edge e, the set of colours assigned to the edges adjacent to e has cardinality either 2 or 4, but not 3. We prove that every bridgeless cubic graph G admits an edge-colouring with 4 colours such that at most 4 5 · |V(G)| edges do not satisfy the above condition. This bound is tight and the Petersen graph is the only connected graph for which the bound cannot be decreased. We obtain such a 4-edge-colouring by using a carefully chosen subset of edges of a perfect matching, and the analysis relies on a simple discharging procedure with essentially no reductions and very few rules.

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Section: (R1)mentioning

confidence: 99%

mentioning

confidence: 99%

Variations on the Petersen Colouring Conjecture

Pirot¹,

Sereni

Škrekovski³

2020

Electron. J. Combin.

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“…This means that the non-3edge-colorable cubic graphs are the main obstacle for proving Conjecture 3. Let us note that in [6] Conjecture 3 is verified for some non-3-edge-colorable bridgeless cubic graphs. Also, in [16] the percentage of edges of a bridgeless cubic graph, which can be made normal in a 5-edge-coloring, is estimated.…”

Section: Normal 5-edge-coloringsmentioning

confidence: 96%

Normal 5-edge-colorings of a family of Loupekhine snarks

Ferrarini

Mazzuoccolo

Mkrtchyan

2020

AKCE International Journal of Graphs and Combinatorics

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In a proper edge-coloring of a cubic graph an edge uv is called poor or rich, if the set of colors of the edges incident to u and v contains exactly three or five colors, respectively. An edge-coloring of a graph is normal, if any edge of the graph is either poor or rich. In this note, we show that some snarks constructed by using a method introduced by Loupekhine admit a normal edge-coloring with five colors. The existence of a Berge-Fulkerson Covering for a part of the snarks considered in this paper was recently proved by Manuel and Shanthi (2015). Since the existence of a normal edge-coloring with five colors implies the existence of a Berge-Fulkerson Covering, our main theorem can be viewed as a generalization of their result.

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“…The importance of H-colourings is mainly due to Jaeger's Conjecture [14] which states that each bridgeless cubic graph G admits a P -colouring (where P is again the Petersen graph). For recent results on P -colourings, known as Petersen-colourings, see for instance [12,13,26,29]. The following proposition shows why we choose to refer to a pair of perfect matchings whose deletion leaves a bipartite subgraph as an S 4 -pair.…”

Section: Statements Equivalent To the S -Conjecturementioning

confidence: 99%

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

Mazzuoccolo

Zerafa²

2020

Ars Math. Contemp.

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In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic graph has three perfect matchings whose intersection is empty. In this paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture originally proposed by the first author, which states that in every simple bridgeless cubic graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges.

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Petersen-colorings and some families of snarks

Cited by 15 publications

References 8 publications

Variations on the Petersen Colouring Conjecture

Variations on the Petersen Colouring Conjecture

Normal 5-edge-colorings of a family of Loupekhine snarks

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

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