On the maximum fraction of edges covered by t perfect matchings in a cubic bridgeless graph

On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks (cyclically 4-edge-connected cubic graph...

The Maximum Number of Edges in a Strongly Multiplicative Graph

A superlinear bound on the number of perfect matchings in cubic bridgeless graphs

Lovász and Plummer conjectured in the 1970’s that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and Seymour in 2008, but in general only linear bounds are known. In this paper, we provide the first superlinear bound in the general case.

Covering a cubic graph by 5 perfect matchings

Berge Conjecture states that every bridgeless cubic graph has 5 perfect matchings such that each edge is contained in at least one of them. In this paper, we show that Berge Conjecture holds for two classes of cubic graphs, cubic graphs with a circuit missing only one vertex and bridgeless cubic graphs with a 2-factor consisting of two circuits. The first part of this result implies that Berge ...

On Perfect Matchings in Matching Covered Graphs

Let G be a matching-covered graph, i.e., every edge is contained in a perfect matching. An edge subsetX ofG is feasible if there exists two perfect matchingsM1 andM2 such that |M1∩X| 6≡ |M2∩X| (mod 2). Lukot’ka and Rollová proved that an edge subset X of a regular bipartite graph is not feasible if and only if X is switching-equivalent to ∅, and they further ask whether a non-feasible set of a ...