发表机构
School of Mathematics and Statistics; Northwestern Polytechnical University（数学与统计学院; 西北工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文否证了Lucchesi和Murty关于简单实心匹配覆盖图最大边数的猜想，证明最大边数为$n^2+2$，并刻画了达到该界的图，同时证明简单实心砖的边数上界为$n^2$。
AI 中文摘要
一个至少有两个顶点的连通图，如果它的每条边都包含在一个完美匹配中，则称为匹配覆盖图。一个匹配覆盖图是实心的，如果它的每个分离割都是紧割。一个没有非平凡紧割的匹配覆盖图，如果是非二分图，则称为砖。每个二分匹配覆盖图都是实心的。Lucchesi和Murty猜想存在一个正整数$N$，使得对于每个整数$n\ge N$，在$2n$个顶点上的简单实心匹配覆盖图的最大边数为$n^2$。在本文中，我们否证了这个猜想，并证明了在$2n$（$n\ge2$）个顶点上的简单实心匹配覆盖图的最大边数为$n^2+2$。此外，我们刻画了达到这个界的图。另外，我们证明了对于$n\ge4$，阶为$2n$的每个简单实心砖至多有$n^2$条边。
英文摘要
A connected graph with at least two vertices is matching covered if each of its edges is contained in a perfect matching. A matching covered graph is solid if every separating cut in it is a tight cut. A matching covered graph which is free of nontrivial tight cuts is a brick if it is nonbipartite. Every bipartite matching covered graph is solid. Lucchesi and Murty conjectured that there exists a positive integer $N$ such that, for every integer $n\ge N$, the maximum number of edges in a simple solid matching covered graph on $2n$ vertices is $n^2$. In this paper, we disprove this Conjecture, and show that the maximum number of edges of a simple solid matching covered graph on $2n$ ($n\ge2$) vertices is $n^2+2$. Moreover, we characterize the graphs attaining this bound. In addition, we prove that every simple solid brick of order $2n$ has at most $n^2$ edges for $n\ge4$.
