Sparsely intersecting perfect matchings in cubic graphs

Abstract

In 1971, Fulkerson made a conjecture that every bridgeless cubic graph contains a family of six perfect matchings such that each edge belongs to exactly two of them; equivalently, such that no three of the matchings have an edge in common. In 1994, Fan and Raspaud proposed a weaker conjecture which requires only three perfect matchings with no edge in common. In this paper we discuss these and other related conjectures and make a step towards Fulkerson’s conjecture by proving the following result: Every bridgeless cubic graph which has a 2-factor with at most two odd circuits contains three perfect matchings with no edge in common.

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Correspondence to Martin Škoviera.

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Máčajová, E., Škoviera, M. Sparsely intersecting perfect matchings in cubic graphs. Combinatorica 34, 61–94 (2014). https://doi.org/10.1007/s00493-014-2550-4

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Mathematics Subject Classification (2000)

  • 05C15
  • 05C70