Topological Circles and Euler Tours in Locally Finite Graphs

We obtain three results concerning the topological cycle space C of a locally finite connected graph G. Confirming a conjecture of Diestel we show that through every edge set E ∈ C there is a topological Euler tour, a continuous map from the circle S 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 (τi)i∈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 (τi). As a corollary we obtain a short proof of one of the fundamental theorems about C, that every E ∈ C is a disjoint union of edge sets of circles in |G|. Third, we show that every set of edges not containing a finite odd cut of G extends to an element of C.