Topological circles and Euler tours in locally finite graphs (Q1028817)

Summary: We obtain three results concerning topological paths ands circles in the end compactification \(|G|\) of a locally finite connected graph \(G\). Confirming a conjecture of Diestel we show that through every edge set \(E\in {\mathcal C}\) there is a topological Euler tour, a continuous map from the circle \(S^1\) to the end compactification \(|G|\) of \(G\) that traverses every edge in \(E\) exactly once and traverses no other edge. Second, we show that for every sequence \((\tau_i)_{i\in \mathbb N}\) of topological \(x\)-\(y\) paths in \(|G|\) there is a topological \(x\)-\(y\) path in \(|G|\) all of whose edges lie eventually in every member of some fixed subsequence of \((\tau_i)\). It is pointed out that this simple fact has several applications some of which reach out of the realm of \(|G|\). Third, we show that every set of edges not containing a finite odd cut of \(G\) extends to an element of \({\mathcal C}\).