Cubic vertices in minimal braces

Abstract

Braces play a fundamental role in matching theory, as they, together with bricks, constitute the basic building blocks of matching covered graphs in the tight cut decomposition. A brace is minimal if deleting any edge from it results in a graph that is not a brace. We prove that every perfect matching of a minimal brace of order at least six contains an edge whose ends both have degree three. Consequently, every such brace has at least three edges with this property. Combining this result with the forest structure induced by the noncubic vertices, we show that every minimal brace $G$ of order $n\ge6$ and size $m$, other than $K_{3,3}$, satisfies $n_3(G)\ge\max\left\{8,\left\lceil\frac{2n+8}{5}\right\rceil, \left\lceil\frac{m-n+4}{2}\right\rceil\right\}$, where $n_3(G)$ is the number of cubic vertices of $G$. Finally, we characterize the graphs attaining the constant lower bound: a minimal brace has exactly eight cubic vertices if and only if it is isomorphic to $B_8$, $B_{10}$, $Q_{10}^{+}$, or $Q_{12}$.