The Cycle Rank Threshold: Perfect Matchings and Property (P) in Bipartite Graphs

We determine the maximum cycle rank for which property~(P) is equivalent to the existence of a perfect matching in bipartite graphs. It is known that this equivalence holds for bipartite graphs of cycle rank at most \(3\)~\cite{puja}. In this paper, we extend the equivalence to bipartite graphs of cycle rank \(4\) and show that this bound is sharp.