Graph-theoretical versus topological ends of graphs
Journal of Combinatorial Theory Series B
Combinatorica
Combinatorica
Topological paths, cycles and spanning trees in infinite graphs
European Journal of Combinatorics - Special issue: Topological graph theory
The Cycle Space of an Infinite Graph
Combinatorics, Probability and Computing
Combinatorics, Probability and Computing
Combinatorics, Probability and Computing
End spaces of graphs are normal
Journal of Combinatorial Theory Series B
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We determine when the topological spaces |G| naturally associated with a graph G and its ends are metrizable or compact. In the most natural topology, |G| is metrizable if and only if G has a normal spanning tree. We give two proofs, one of them based on Stone's theorem that metric spaces are paracompact. We show that |G| is compact in the most natural topology if and only if no finite vertex separator of G leaves infinitely many components. When G is countable and connected, this is equivalent to the existence of a locally finite spanning tree. The proof uses ultrafilters and a lemma relating ends to directions.