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The Growth of Infinite Graphs: Boundedness and Finite Spreading

Published online by Cambridge University Press:  12 September 2008

Reinhard Diestel  and
Imre Leader
Reinhard Diestel
Affiliation:
Faculty of Mathematics (SFB 343), Bielefeld University, D-4800 Bielefeld, Germany
Imre Leader
Affiliation:
Department of Pure Mathematics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 1SB, England
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Abstract

An infinite graph is called bounded if for every labelling of its vertices with natural numbers there exists a sequence of natural numbers which eventually exceeds the labelling along any ray in the graph. Thomassen has conjectured that a countable graph is bounded if and only if its edges can be oriented, possibly both ways, so that every vertex has finite out-degree and every ray has a forward oriented tail. We present a counterexample to this conjecture.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 1 , March 1994 , pp. 51 - 55
Copyright
Copyright © Cambridge University Press 1994

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References

[1]Diestel, R. and Leader, I. (1992) A proof of the bounded graph conjecture. Invent. Math. 108 131–162.CrossRefGoogle Scholar
[2]Halin, R. (1989) Some problems and results in infinite graphs. In: Andersen, L. D. et al. , (eds.) Graph Theory in Memory of G. A. Dirac. Annals of Discrete Mathematics 41.Google Scholar
[3]Halin, R. (1992) Bounded graphs. In: Diestel, R. (ed.) Directions in infinite graph theory and combinatorics. Topics in Discrete Mathematics 3.Google Scholar
[4]Rado, R. (1964) Universal graphs and universal functions. Acta Arith. 9 331–340.CrossRefGoogle Scholar
[5]Thomassen, C. (private communication).Google Scholar