Topological Circles and Euler Tours in Locally Finite Graphs

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 $E\in {\cal C}$ 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 $(\tau_i)_{i\in \Bbb N}$ 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 $\cal C$.