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Matchings in Arbitrary Graphs

  • R. A. Brualdi (a1)

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1. Tutte(10) has given necessary and sufficient conditions in order that a finite graph have a perfect matching. A different proof was given by Gallai(4). Berge(1) (and Ore (7)) generalized Tutte's result by determining the maximum cardinality of a matching in a finite graph. In his original proof Tutte used the method of skew symmetric determinants (or pfaffians) while Gallai and Berge used the much exploited method of alternating paths. Another proof of Berge's theorem, along with an efficient algorithm for constructing a matching of maximum cardinality, was given by Edmonds (2). In another paper (12) Tutte extended his conditions for a perfect matching to locally finite graphs.

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(1)Berge, C.The theory of graphs. Methuen (London) and J. Wiley and Sons (New York), 1962.
(2)Edmonds, J.Paths, trees and flowers. Canad. J. Math. 17 (1965), 449467.
(3)Edmonds, J. and Fulkerson, D. R.Transversals and matroid partition. J. Res. Nat. Bur. Standards Sect. B 69 (1965), 147153.
(4)Gallai, T.On factorisations of graphs. Acta Math. Acad. Sci. Hungar. 1 (1950), 133152.
(5)Gottschalk, W. H.Choice functions and Tychonoff's theorem. Proc. Amer. Math. Soc. 2 (1951), 172.
(6)Hall, P.On representatives of subsets. J. London Math. Soc. 10 (1935), 2630.
(7)Ore, O.Graphs and subgraphs. Trans. Amer. Math. Soc. 84 (1957), 109137.
(8)Rado, R.Axiomatic treatment of rank in infinite sets. Canad. J. Math. 1 (1949), 337343.
(9)Rado, R.Note on the transfinite case of Hall's theorem on representatives. J. London Math. Soc. 42 (1967), 321324.
(10)Tutte, W. T.The factorization of linear graphs. J. London Math. Soc. 22 (1947), 107111.
(11)Tutte, W. T.The factors of graphs. Canad. J. Math. 4 (1952), 314328.
(12)Tutte, W. T.The factorization of locally finite graphs. Canad. J. Math. 2 (1950), 4449.
(13)Whitney, H.On the abstract properties of linear dependence. Amer. J. Math. 57 (1935), 509533.

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