Compact compatible topologies for graphs with small cycles

Victor Neumann-Lara and Richard G. Wilson

International Journal of Mathematics and Mathematical Sciences, 2005, vol. 2005, 1-11

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.

Date: 2005
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:190408

DOI: 10.1155/IJMMS.2005.2195

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