Deciding whether four perfect matchings can cover the edges of a snark is NP-complete

Abstract

A conjecture of Berge predicts that every bridgeless cubic graph can have its edges covered with at most five perfect matchings. If the graph in question has no 3-edge-colouring, then at least four perfect matchings are necessary. It was proved by Esperet and Mazzuoccolo (2014) [3] that it is NP-complete to decide whether four perfect matchings are enough to cover the edges of a bridgeless cubic graph. A disadvantage of the proof (noted by the authors) is that the constructed graphs have 2-cuts. In this paper we show that small cuts can be altogether avoided and that the problem remains NP-complete even for nontrivial snarks – that is, cyclically 4-edge-connected cubic graphs with no 3-edge-colouring. As a by-product, we provide a rich family of nontrivial snarks that cannot be covered with four perfect matchings. The methods rely on the theory of tetrahedral flows developed in Máčajová and Škoviera (2021) [9].