Ends in spanning trees

✍ Scribed by Xingxing Yu

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 97 KB
• Volume: 104
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)90455-o

✦ Synopsis

We refer to for terminology not specified here. Graphs mentioned in this note are undirected, simple. The following definition is due to Halin [l]: an end E of an infinite graph G is a set of l-way infinite paths in G such that P, Q E 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 l-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 l-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.