Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T18:08:36.712Z Has data issue: false hasContentIssue false
  • English
  • Français

A Characterisation of Strict Matching Matroids

Published online by Cambridge University Press:  12 September 2008

Victor Bryant
Victor Bryant
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England
Get access Rights & Permissions [Opens in a new window]

Abstract

It is well known that a matroid is a transversal matroid if and only if it is a matching matroid (in the sense that it is the restriction of the matching structure of some graph to a subset of its vertices). A simple proof of that result is now known and in this paper it is used to answer the long-standing question of which transversal matroids are “strict” matching matroids; i.e. actually equal to the matching structure of a graph. We develop a straightforward test of “coloop-surfeit” that can be applied to any transversal matroid, and our main theorem shows that a transversal matroid is a strict matching matroid if and only if it has even rank and coloop-surfeit. Furthermore, the proofs are algorithmic and enable the construction of an appropriate graph from any presentation of a strict matching matroid.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 1 , Issue 3 , September 1992 , pp. 209 - 217
Copyright
Copyright © Cambridge University Press 1992

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

[1]Bondy, J. A. and Welsh, D. J. A. (1971) Some results on transversal matroids and constructions for identically self-dual matroids, Quart. J. Math., 22 435–451.CrossRefGoogle Scholar
[2]Brualdi, R. A. and Scrimger, E. B. (1968) Exchange Systems, matchings and transversals, J. Comb. Th., 5 244–257.CrossRefGoogle Scholar
[3]Edmonds, J. (1965) Paths, trees and flowers, Canad. J. Math., 17 449–467.CrossRefGoogle Scholar
[4]Edmonds, J. and Fulkerson, D. R. (1965) Transversals and matroid partition, J. Res. Nat. Bur. Stand., 69B 147–153.CrossRefGoogle Scholar
[5]Gallai, T. (1963) Neuer Beweis eines Tutteschen Satzes, Magyar Tud. Akad. Mat. Kutató Int. KözL, 8 135–139.Google Scholar
[6]Lovász, L. (1979) Combinatorial problems and exercises, North Holland, Amsterdam.Google Scholar
[7]Mirsky, L. and Perfect, H. (1967) Applications of the notion of independence to combinatorial analysis, J.Comb. Th., 2 327–357.CrossRefGoogle Scholar
[8]Triesch, E. (1992) A short proof that matching matroids are transversal, App. Math. Letters, 5.2 19–20.CrossRefGoogle Scholar
[9]Welsh, D. J. A. (1976) Matroid theory, Academic Press, London.Google Scholar