International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 14, Pages 2195-2205
doi: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 Victor Neumann-Lara and Richard G. Wilson. 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.