Unions of perfect matchings in r-graphs

An r-regular graph is said to be an r-graph if |∂(X)|≥r for each odd set X⊆V(G), where |∂(X)| denotes the set of edges with precisely one end in X. Note that every connected bridgeless cubic graph is a 3-graph. The Berge Conjecture states that every 3-graph G has five perfect matchings such that each edge of G is contained in at least one of them. Likewise, generalization of the Berge Conjecture asserts that every r-graph G has 2r−1 perfect matchings that covers each e∈E(G) at least once. A natural question to ask in the light of the Generalized Berge Conjecture is that what can we say about the proportion of edges of an r-graph that can be covered by union of t perfect matchings? In this paper we provide a lower bound to this question. We will also present a new conjecture that might help towards the proof of the Generalized Berge Conjecture.