Let \(G\) be a cubic bipartite plane graph that has a perfect matching. If \(M\) is any perfect matching of \(G\), then \(G\) has a face that is \(M\)-alternating.If \(f\) is any face of \(G\), then there is a perfect matching \(M\) such that \(f\) is \(M\)-alternating.There is a simple algorithm for visiting all perfect matchings of \(G\) beginning at one.
There are infinitely many cubic plane graphs that have perfect matchings but whose matching transformation graphs are completely disconnected.
Several problems are proposed.
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