On a Conjecture of Lov asz Concerning

In 1987, Lovv asz conjectured that every brick G diierent from K 4 , C 6 , and the Petersen graph has an edge e such that G e is a matching covered graph with exactly one brick. Lovv asz and Vempala announced a proof of this conjecture in 1994. Their paper is under preparation. We present here an independent proof of their theorem. We shall in fact prove that if G is any brick diierent from K 4 and C 6 and does not have the Petersen graph as its underlying simple graph, then it has an edge e such that G e is a matching covered graph with exactly one brick, with the additional property that the underlying simple graph of that one brick is diierent from the Petersen graph. Our proof involves establishing an interesting new property of the Petersen graph.