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.