Ore, Berge–Tutte, and Gallai–Edmonds

We present a short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem based on Ore’s Defect Formula and Anderson’s proof of Tutte’s 1-Factor Theorem from Hall’s Theorem. The fundamental theorems on matchings in graphs have been proved in many ways, and generally they all imply each other. The most well-known is Hall’s Theorem [7] characterizing when a bipartite graph has a matching that covers one partite set. Anderson [1] used Hall’s Theorem to prove Tutte’s 1-Factor Theorem [10], characterizing when a graph has a perfect matching. Meanwhile, Ore [8] extended Hall’s Theorem to give a min-max formula for the maximum size of a matching in a bipartite graph. Similarly, Berge [2] extended Tutte’s 1-Factor Theorem to give a min-max formula (known as the Berge–Tutte Formula) for the maximum size of a matching in a general graph. We show that Anderson’s approach proves the Berge–Tutte Formula from Ore’s Defect Formula as easily as it proves Tutte’s 1-Factor Theorem from Hall’s Theorem. The same approach then yields the Gallai–Edmonds Structure Theorem [3, 5, 6], which describes all the maximum matchings in a given graph. This is shorter than earlier proofs by not needing a characterization of factor-critical graphs or a “Stability Lemma”. For a set of vertices S in a graph G, let NG(S) or N(S) denote the set of vertices having at least one neighbor in S. An X,Y -bigraph is a bipartite graph with partite sets X and Y . A matching is a set of pairwise non-incident edges. An obvious necessary condition for the existence of a matching that covers X in an X,Y -bigraph is that |N(S)| ≥ |S| for all S ⊆ X. This is Hall’s Condition, and Hall’s Theorem states that it is also sufficient. The defect df(S) of a set S ⊆ X in an X,Y -bigraph is |N(S)|−|S|. The matching number α(G) is the maximum size of a matching in G. By applying Hall’s Theorem to the graph obtained from an X,Y -bigraph G by adding max{df(S)} vertices to Y that are adjacent to all of X, Ore [8] observed that α(G) = |X| −max{df(S)}. ∗Department of Mathematics, University of Illinois, Urbana, IL 61801, [email protected]. This research is partially supported by the National Security Agency under Award No. H98230-06-1-0065.