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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
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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

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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 Unpublished.Google Scholar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