On a Norine–Thomas conjecture concerning minimal bricks

Abstract

A 3‐connected graph is a brick if has a perfect matching, for each pair of vertices of . A brick is minimal if ceases to be a brick for every edge . Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists such that every minimal brick has at least cubic vertices. In this paper, we prove this conjecture holds for all minimal bricks of an average degree no less than 23/5. As its corollary, we show that each minimal brick on vertices contains more than vertices of degree at most four.