3-lines in minimal braces

Prove that every minimal brace—that is, a bipartite matching covered graph with no nontrivial tight cuts that is minimal with respect to the matching covered property—on at least six vertices contains a 3-line, meaning an edge whose two endpoints both have degree 3.

Background

In this paper, a brace is a bipartite matching covered graph with no nontrivial tight cuts, and a minimal brace is a brace that ceases to be matching covered upon deletion of any edge. An edge is called a 3-line if both its endpoints have degree 3.

Prior work shows structural richness of minimal braces: Lou (1999) proved that any minimal brace with at least six vertices has minimum degree 3 and at least ceil((2|V|+2)/5) cubic vertices. Motivated by their results on minimal matching covered graphs (including the existence of multiple 3-lines when the minimum degree is at least 3), the authors propose the following conjecture specific to minimal braces.

References

In Section 6, we present a conjecture that states that each minimal brace (see its definition in Sections 4 and 6) with at least 6 vertices has a 3-line.

— Adjacent vertices of small degree in minimal matching covered graphs  (2604.00361 - He et al., 1 Apr 2026) in Section 1 (Introduction)