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International Journal of Mathematics and Mathematical Sciences
Volume 2005, Issue 14, Pages 2195-2205
http://dx.doi.org/10.1155/IJMMS.2005.2195

Compact compatible topologies for graphs with small cycles

1Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, México DF 04510, Mexico
2Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenida San Rafael Atlixco 186, Apartado Postal 55-532, México DF 09340, Mexico

Received 11 November 2004; Revised 12 July 2005

Copyright © 2005 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A topology τ on the vertices of a comparability graph G is said to be compatible with G if each subgraph H of G is graph-connected if and only if it is a connected subspace of (G,τ). In two previous papers we considered the problem of finding compatible topologies for a given comparability graph and we proved that the nonexistence of infinite paths was a necessary and sufficient condition for the existence of a compact compatible topology on a tree (that is to say, a connected graph without cycles) and we asked whether this condition characterized the existence of a compact compatible topology on a comparability graph in which all cycles are of length at most n. Here we prove an extension of the above-mentioned theorem to graphs whose cycles are all of length at most five and we show that this is the best possible result by exhibiting a comparability graph in which all cycles are of length 6, with no infinite paths, but which has no compact compatible topology.