发表机构
Indian Institute of Technology Madras（印度马德拉斯理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定二分图中使性质(P)等价于存在完美匹配的最大圈秩，将该等价关系从圈秩≤3推广到圈秩为4并证明其界是紧的。
AI 中文摘要
我们确定二分图中使性质(P)等价于存在完美匹配的最大圈秩。已知该等价关系对圈秩不超过3的二分图成立[1]，本文将其推广到圈秩为4的二分图，并证明该界是紧的。
英文摘要
A graph on \(n\) vertices is called a Parter graph if there exists a nonsingular symmetric matrix, whose nonzero off-diagonal entries correspond exactly to the edges of the graph, such that all of its principal submatrices of order \(n-1\) are singular. Previously, a graph satisfying this condition was said to have property~(P). It was proved that, for bipartite graphs of cycle rank at most \(3\), being a Parter graph is equivalent to the existence of a perfect matching. We extend this result to cycle rank \(4\), proving that every bipartite graph of cycle rank at most \(4\) is a Parter graph if and only if it has a perfect matching. Furthermore, we show that this bound is sharp by constructing, for every integer \(r\ge5\), a connected balanced bipartite Parter graph of cycle rank \(r\) that has no perfect matching.
CommentsTitle and abstract updated. Terminology changed from "property (P)" to "Parter graph" to align with standard literature. Added new results
