The Cycle Rank Threshold: Perfect Matchings and Bipartite Parter Graphs

Abstract

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.

Title and abstract updated. Terminology changed from "property (P)" to "Parter graph" to align with standard literature. Added new results