A short proof of the Berge-Tutte Formula and the Gallai-Edmonds Structure Theorem

We present a short proof of the Berge–Tutte Formula and the Gallai–Edmonds Structure Theorem from Hall’s Theorem. The fundamental theorems on matchings in graphs have been proved in many ways. The most famous of these results is Hall’s Theorem [6], characterizing when a bipartite graph has a matching that covers one partite set. Anderson [1] used Hall’s Theorem to prove Tutte’s 1Factor Theorem [9], characterizing when a graph has a perfect matching. Berge [2] extended Tutte’s 1-Factor Theorem to a min-max formula (known as the Berge–Tutte Formula) for the maximum size of a matching in a general graph. In fact, Anderson’s approach proves the Berge–Tutte Formula as easily as it proves Tutte’s 1-Factor Theorem. Using the Berge–Tutte Formula, it then also yields the Gallai–Edmonds Structure Theorem [3, 4, 5], which describes all the maximum matchings in a given graph. Our proof by this method is shorter than earlier inductive proofs (see Theorem 3.2.1 of [8], for example) by not needing a characterization of factor-critical graphs or a “Stability Lemma” (Lemma 3.2.2 in [8]). We are indebted to the referee for pointing out the paper by Kotlov [7], which gives another short proof along similar lines to that given here. For a set S of vertices 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. In an X,Y -bigraph, an obvious necessary condition for a matching that covers X is that |N(S)| ≥ |S| for all S ⊆ X. This is Hall’s Condition, and Hall’s Theorem [6] states that it is also sufficient. A 1-factor is a spanning 1-regular subgraph; its edge set is a perfect matching. In a graph H, let o(H) be the number of odd components (those having an odd number of vertices). In a graph G, an obvious necessary condition for a 1-factor is that o(G− S) ≤ |S| whenever S ⊆ V (G). This is Tutte’s Condition; Tutte proved that it is also sufficient. In a graph G, the deficiency defG(S) or def(S) is o(G−S)−|S|. Covering all vertices in an odd component of G−S by a matching in G requires matching one of its vertices with a vertex ∗Department of Mathematics, University of Illinois, Urbana, IL 61801, [email protected]. This research is partially supported by the National Security Agency under Award No. H98230-10-1-0363.