Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-26T07:45:03.975Z Has data issue: false hasContentIssue false
  • English
  • Français

Matchings in Arbitrary Graphs

Published online by Cambridge University Press:  24 October 2008

R. A. Brualdi
R. A. Brualdi
Affiliation:
University of Sheffield, Sheffield, England and University of Wisconsin, Madison, Wisconsin, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 69 , Issue 3 , May 1971 , pp. 401 - 407
Copyright
Copyright © Cambridge Philosophical Society 1971

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Berge, C.The theory of graphs. Methuen (London) and J. Wiley and Sons (New York), 1962.Google Scholar
(2)Edmonds, J.Paths, trees and flowers. Canad. J. Math. 17 (1965), 449–467.CrossRefGoogle Scholar
(3)Edmonds, J. and Fulkerson, D. R.Transversals and matroid partition. J. Res. Nat. Bur. Standards Sect. B 69 (1965), 147–153.Google Scholar
(4)Gallai, T.On factorisations of graphs. Acta Math. Acad. Sci. Hungar. 1 (1950), 133–152.CrossRefGoogle Scholar
(5)Gottschalk, W. H.Choice functions and Tychonoff's theorem. Proc. Amer. Math. Soc. 2 (1951), 172.Google Scholar
(6)Hall, P.On representatives of subsets. J. London Math. Soc. 10 (1935), 26–30.CrossRefGoogle Scholar
(7)Ore, O.Graphs and subgraphs. Trans. Amer. Math. Soc. 84 (1957), 109–137.Google Scholar
(8)Rado, R.Axiomatic treatment of rank in infinite sets. Canad. J. Math. 1 (1949), 337–343.CrossRefGoogle Scholar
(9)Rado, R.Note on the transfinite case of Hall's theorem on representatives. J. London Math. Soc. 42 (1967), 321–324.CrossRefGoogle Scholar
(10)Tutte, W. T.The factorization of linear graphs. J. London Math. Soc. 22 (1947), 107–111.Google Scholar
(11)Tutte, W. T.The factors of graphs. Canad. J. Math. 4 (1952), 314–328.CrossRefGoogle Scholar
(12)Tutte, W. T.The factorization of locally finite graphs. Canad. J. Math. 2 (1950), 44–49.CrossRefGoogle Scholar
(13)Whitney, H.On the abstract properties of linear dependence. Amer. J. Math. 57 (1935), 509–533.CrossRefGoogle Scholar