Compact compatible topologies for graphs with small cycles

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.