Linear degree-three-vertex conjecture for minimal bricks

A brick is a graph satisfying the usual matching-theoretic brick conditions, and a minimal brick is a brick minimal with respect to the relevant brick-preserving reduction. For a graph $G$, write $V(G)$ for its vertex set and let $|V(G)|$ denote its number of vertices.

Linear degree-three-vertex conjecture. There exists $\alpha > 0$ such that every minimal brick $G$ has at least

$$\alpha |V(G)|$$

vertices of degree three.

This conjecture strengthens the preceding result that every minimal brick has at least three vertices of degree three. Even the weaker assertion that every brick has at least four vertices of degree three is described as requiring new ideas or a substantial refinement of the authors' techniques.