On a Norine–Thomas conjecture concerning minimal bricks

Abstract

A 3-connected graph $G$ is a brick if $G-S$ has a perfect matching, for each pair $S$ of vertices of $G$. A brick $G$ is minimal if $G-e$ ceases to be a brick for every edge $e\in E\left(G\right)$. Norine and Thomas proved that each minimal brick contains at least three vertices of degree three and made a stronger conjecture: there exists $\alpha >0$ such that every minimal brick $G$ has at least $\alpha \mid V\left(G\right)\mid$ 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 $n$ vertices contains more than $n/5$ vertices of degree at most four.