Bases for the Matching Lattice of Matching Covered Graphs

Abstract

This article contains a brief introduction to the theory of matching covered graphs: ear-decompositions, the matching lattice and the most important results of the theory. In its last section we prove that the matching lattice of any matching covered graph has a basis consisting solely of perfect matchings and we present a conjecture relating the minimum number of double ears of any eardecomposition of a matching covered graph and the number of bricks and bricks isomorphic to the Petersen graph in any brick decomposition of the same graph.