The cubic vertices of solid minimal bricks

A 3-connected graph is a brick if, for any two vertices u and v, G−u−v has a perfect matching. A brick G is minimal if G−e is not a brick for every edge e of G. Norine and Thomas [J. Combin. Theory Ser. B, 96(4) (2006), pp. 505-513.] conjectured that there exists α>0 such that every minimal brick G contains at least α|V(G)| cubic vertices. A brick is solid if for any two disjoint odd cycles C1 and C2, G−V(C1∪C2) has no perfect matching. In this paper, by proving that every solid minimal brick G has at least 2|V(G)|/5 cubic vertices, we obtain the first non-trivial class of graphs that support the above conjecture.