TRACTATUS on disjoint matchings in cubic graphs

In early 70s Berge conjectured that any bridgeless cubic graph contains five perfect matchings such that each edge belongs to at least one of them. In 1972 Fulkerson conjectured that, in fact, we can find six perfect matchings containing each edge exactly twice. By introducing the concept of an r-graph (a remarkable generalization of one of bridgeless cubic graph) Seymour in 1979 conjectured that every r-graph contains 2r perfect matchings such that each edge belongs to exactly two of them. We investigate the following problem: what is the percentage of edges of a cubic graph that we may hope to cover by its k matchings. The investigation of this problem in the class of bridgeless cubic graphs has led us to new conjectures and interesting examples of cubic graphs.