The maximum size of simple solid matching covered graphs

A connected graph with at least two vertices is matching covered if each of its edges is contained in a perfect matching. A matching covered graph is solid if every separating cut in it is a tight cut. A matching covered graph which is free of nontrivial tight cuts is a brick if it is nonbipartite. Every bipartite matching covered graph is solid. Lucchesi and Murty conjectured that there exists a positive integer $N$ such that, for every integer $n\ge N$, the maximum number of edges in a simple solid matching covered graph on $2n$ vertices is $n^2$. In this paper, we disprove this Conjecture, and show that the maximum number of edges of a simple solid matching covered graph on $2n$ ($n\ge 2$) vertices is $n^2+2$. Moreover, we characterize the graphs attaining this bound. In addition, we prove that every simple solid brick of order $2n$ has at most $n^2$ edges for $n\ge 4$.