A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 we call such a collection of perfect matchings a Berge covering of G. A cubic graph G is called a Kotzig graph if G has a 3-edge-coloring such that each pair of colors forms a hamiltonian circuit introduced by R. Haggkvist, K. Markstrom, J Combin Theory Ser B 96 2006, 183-206. In this article, we prove that if there is a vertex w of a cubic graph G such that G-wi¾?, the graph obtained from G-w by suppressing all degree two vertices is a Kotzig graph, then G has a Berge covering. We also obtain some results concerning the so-called 5-even subgraph double cover conjecture.

On Perfect Matching Coverings and Even Subgraph Coverings

If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find $6$ perfect matchings (a {\em 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 {\em 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.

https://hal.archives-ouvertes.fr/hal-00392009/document

On Fulkerson conjecture

Weare pleased to present this special issue ofDiscrete AppliedMathematics, which is devoted to the 12th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2013), held at the University of Twente, Enschede, The Netherlands, between May 21 and May 23, 2013. More than 80 scientists from all over the world attended the workshop, and we take this opportunity to thank all CTW 2013 participants for their presentations and lively discussions at the workshop. After the workshop, we invited the participants and the community at large to submit manuscripts with research results related to the themes and topics of the workshop. From the submitted manuscripts, we have selected, after a careful refereeing process, twelve articles to be included in this special issue. We thank the reviewers for their insightful reports and the authors for the quality of their contributions. Special thanks are due to Endre Boros, Editor-in-Chief, who accepted to host this issue dedicated to CTW 2013 in Discrete Applied Mathematics, thereby following a long tradition. We also thank Katie D’Agosta, Publications Manager, for her help and valuable support. The CTW started as a series of biennial meetings at the Universities of Cologne in Germany and Twente in the Netherlands. Ever-increasing interest has turned the CTW into a now annual event with several other universities as additional partners. The scope of the workshop comprises the theory and applications of discrete algorithms, graphs, and combinatorial structures in a broad sense. We hope for a continuing interest in these CTW workshops and are looking forward to contributions in the future from all interested researchers.

/pdf/proceedings-of-the-12th-cologne-twente-workshop-on-graphs-5bl0cq185y.pdf

Proceedings of the 12th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2013)

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic (3-valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non 3-edge-colorable cubic graph. In contrast to Tutte’s 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5-edge-connected.

Reduction of the Berge-Fulkerson conjecture to cyclically 5-edge-connected snarks

A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. The Berge–Fulkerson conjecture proposed in 1971 states that every bridgeless cubic graph contains a family of six perfect matchings such that each edge is contained in exactly two of them. This conjecture holds trivialy for 3-edge colorable graphs. Thus a possible minimum counterexample for the conjecture is a snark. In this paper we shown that the conjecture holds for families of snarks such as Loupekhine snarks of first and second kind and the Watkins snark. We also determine the excessive index for Loupekhine snarks of first and second kind.

Berge–Fulkerson Conjecture on Certain Snarks

http://www.math.wvu.edu/~cqzhang/Publication-files/my-paper/DM-2009-Fulkerson.pdf

A note on Berge-Fulkerson coloring

Nowhere-zero 4-flow in almost Petersen-minor free graphs

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A note on Berge-Fulkerson coloring

Nowhere-zero 4-flow in almost Petersen-minor free graphs