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If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6perfect matchings $M_1,...,M_6$ of $G$ with the property that every edge of $G$is contained in exactly two of them and Berge conjectured that its edge set canbe covered by 5 perfect matchings. We define $\tau(G)$ as the least number ofperfect matchings allowing to cover the edge set of a bridgeless cubic graphand we study this parameter. The set of graphs with perfect matching index 4seems interesting and we give some informations on this class.

On the perfect matching index of bridgeless cubic graphs