发表机构
School of Mathematics and Statistics, North China University of Water Resources and Electric Power; School of Mathematics and Statistics, Zhengzhou University（华北水利水电大学数学与统计学院; 郑州大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明极小括号图的每个完美匹配含两端度均为三的边，并给出三次顶点数的下界，刻画了达到常数下界的图。
AI 中文摘要
括号图在匹配理论中扮演着基础性角色，因为它们与砖图一起，在紧割分解中构成了匹配覆盖图的基本构建块。若从图中删除任意一条边后得到的图不再是括号图，则该括号图称为极小的。我们证明，阶数至少为六的极小括号图的每个完美匹配都包含一条两端点度数均为三的边。因此，每个这样的括号图至少有三条具有此性质的边。将此结果与非三次顶点诱导的森林结构相结合，我们证明每个阶数$n\ge6$、大小$m$的极小括号图$G$（除$K_{3,3}$外）满足$n_3(G)\ge\max\left\{8,\left\lceil\frac{2n+8}{5}\right\rceil, \left\lceil\frac{m-n+4}{2}\right\rceil\right\}$，其中$n_3(G)$是$G$的三次顶点数。最后，我们刻画了达到常数下界的图：一个极小括号图恰好有八个三次顶点当且仅当它同构于$B_8$、$B_{10}$、$Q_{10}^{+}$或$Q_{12}$。
英文摘要
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 $Q_{12}$.
