Ends in spanning trees

Abstract We refer to [2] for terminology not specified here. Graphs mentioned in this note are undirected, simple. The following definition is due to Halin [1]: an end E of an infinite graph G is a set of 1-way infinite paths in G such that P,Q ϵ E iff for any finite subset R of V(G) there is a finite path in G −R joining P and Q. An end E of G is free if there is an finite subset R of the vertex set V(G) such that G − R has a component whose 1-way infinite paths belong to E. Zelinka [3] proved that any spanning tree of a connected infinite locally finite graph G contains at least one 1-way infinite path from any free end of G, and he conjectured that the word ‘free’ can be dropped. In this note, we prove the following stronger result.