Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-28T13:10:49.316Z Has data issue: false hasContentIssue false
  • English
  • Français

A counter-example to ‘Wagner's conjecture’ for infinite graphs

Published online by Cambridge University Press:  24 October 2008

Robin Thomas
Robin Thomas
Affiliation:
Department of Mathematical Analysis, Charles University, Sokolovská 83, 186 00 Prague 8, Czechoslovakia
Get access Rights & Permissions [Opens in a new window]

Extract

Wagner made the conjecture that given an infinite sequence G1, G2, … of finite graphs there are indices i < j such that Gi is a minor of Gj. (A graph is a minor of another if the first can be obtained by contraction from a subgraph of the second.) The importance of this conjecture is that it yields excluded minor theorems in graph theory, where by an excluded minor theorem we mean a result asserting that a graph possesses a specified property if and only if none of its minors belongs to a finite list of ‘forbidden minors’. A widely known example of an excluded minor theorem is Kuratowski's famous theorem on planar graphs; one of its formulations says that a graph is planar if and only if it has neither K5 nor K3, 3 as a minor. But several other excluded minor theorems have been discovered by now (see e.g. [7–9]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 103 , Issue 1 , January 1988 , pp. 55 - 57
Copyright
Copyright © Cambridge Philosophical Society 1988

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]Galvin, F.. Unpublished.Google Scholar
[2]Kuratowski, K.. Topologie, vol. I. Monografie Matematyczne (PWN, Warszawa, 1952).Google Scholar
[3]Nash-Williams, C. St. J. A.. On well-quasi-ordering infinite trees. Proc. Cambridge Philos. Soc. 61 (1965), 697–720.CrossRefGoogle Scholar
[4]Robertson, N. and Seymour, P. D.. Graph minors I–XVIII, preprints.Google Scholar
[5]Simon, P.. Private communication.Google Scholar
[6]Thomas, R.. Well-quasi-ordering infinite graphs with forbidden finite planar minor. In preparation.Google Scholar
[7]Wagner, K.. Über eine Erweiterung des Satzes von Kuratowski. Deutsche Math. 2 (1937), 280–285.Google Scholar
[8]Wagner, K.. Bemerkungen zu Hadwigers Vermutung. Math. Ann. 141 (1960), 433–451.CrossRefGoogle Scholar
[9]Wagner, K.. Über eine Eigenschaft der ebenen Komplexe. Math. Ann. 114 (1937), 570–590.CrossRefGoogle Scholar