Ear Decomposition of Factor-Critical Graphs and Number of Maximum Matchings

Abstract

A connected graph \(G\) is said to be factor-critical if \(G-v\) has a perfect matching for every vertex \(v\) of \(G\). Lovasz proved that every factor-critical graph has an ear decomposition. In this paper, the ear decomposition of the factor-critical graphs \(G\) satisfying that \(G-v\) has a unique perfect matching for any vertex \(v\) of \(G\) with degree at least 3 is characterized. From this, the number of maximum matchings of factor-critical graphs with the special ear decomposition is obtained.