Geodesic
On Fulkerson conjecture
Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 253-272

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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.
Keywords: cubic graph, perfect matchings 
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Fouquet, Jean-Luc; Vanherpe, Jean-Marie. On Fulkerson conjecture. Discussiones Mathematicae. Graph Theory, Tome 31 (2011) no. 2, pp. 253-272. http://geodesic.mathdoc.fr/item/DMGT_2011_31_2_a3/