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 2 r − 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. [ABSTRACT FROM AUTHOR]