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RESIDUAL PROPERTIES OF SIMPLE GRAPHS

Part of: Graph theory Low-dimensional topology

Published online by Cambridge University Press:  18 August 2010

BELINDA TROTTA
BELINDA TROTTA*
Affiliation:
Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia (email: B.Trotta@mmassociates.com.au)
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Abstract

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Clark et al. [‘The axiomatizability of topological prevarieties’, Adv. Math.218 (2008), 1604–1653] have shown that, for k≥2, there exists a Boolean topological graph that is k-colourable but not topologically k-colourable; that is, for every ϵ>0, it cannot be coloured by a paintbrush of width ϵ. We generalize this result to show that, for k≥2, there is a Boolean topological graph that is 2-colourable but not topologically k-colourable. This graph is an inverse limit of finite graphs which are shown to exist by an Erdős-style probabilistic argument of Hell and Nešetřil [‘The core of a graph’, Discrete Math.109 (1992), 117–126]. We use the fact that there exists a Boolean topological graph that is 2-colourable but not k-colourable, and some other results (some new and some previously known), to answer the question of which finitely generated topological residual classes of graphs are axiomatizable by universal Horn sentences. A more general version of this question was raised in the above-mentioned paper by Clark et al., and has been investigated by various authors for other structures.

Keywords

random graphgraph colouringBoolean topological graphuniversal Horn classaxiomatization

MSC classification

Secondary: 05C80: Random graphs 05C15: Coloring of graphs and hypergraphs 57M15: Relations with graph theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 3 , December 2010 , pp. 488 - 504
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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