We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S¹ in the graph compactified by its ends. We prove that this cycle space consists of precisely the sets of edges that meet every finite cut evenly, and that the spanning trees whose fundamental cycles generate this cycle space are precisely the end-faithful spanning trees. We also generalize Euler's theorem by showing that a locally finite connected graph with ends contains a closed topological curve traversing every edge exactly once if and only if its entire edge set lies in this cycle space.

We adapt the cycle space of a finite or locally graph to graphs with vertices of infinite degree, using as cycles the homeomorphic images of the unit circle S in the graph together with its ends. We characterize the spanning trees whose fundamental cycles generate this cycle space, and prove infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decomposition into single cycles and orthogonality to cuts.

Abstract: We develop a general model of edge spaces in order to generalize, unify, and simplify previous work on cycle spaces of infinite graphs. We give simple topological criteria to show that the fundamental cycles of a (generalization of a) spanning tree generate the cycle space in a connected, compact, weakly Hausdorff edge space. Furthermore, in such a space, the orthogonal complement of the bond space is the cycle space. This work unifies the two different notions of cycle space as introduced by Diestel

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 ∈ 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 (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). 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 C.

This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open

Abstract Bonnington and Richter defined the cycle space of an infinite graph to consist of the sets of edges of subgraphs having even degree at every vertex. Diestel and Kühn introduced a different cycle space of infinite graphs based on allowing infinite circuits. A more general point of view was taken by Vella and Richter, thereby unifying these cycle spaces. In particular, different compactifications of locally finite graphs yield different topological spaces that have different cycle spaces. In this work, the Vella-Richter approach is pursued by considering cycle spaces over all fields, not just Z2. In order to understand “orthogonality” relations, it is helpful to consider two different cycle spaces and three different bond spaces. We give an analogue of the “edge tripartition theorem ” of Rosenstiehl and Read and show that the cycle spaces of different compactifications of a locally finite graph are related. 1

We investigate the end spaces of infinite dual graphs. We show that there exists a natural homeomorphism between the end spaces of a graph and its dual, and that this homeomorphism maps thick ends to thick ends. Along the way, we prove that Tutte-connectivity is invariant under taking (infinite) duals.

A circle in a graph G is a homeomorphic image of the unit circle in the Freudenthal compactification of G, a topological space formed from G and the ends of G. Bruhn conjectured that every locally finite 4-connected planar graph G admits a Hamilton circle, a circle containing all points in the Freudenthal compactification of G that are vertices and ends of G. We prove this conjecture for graphs with no dividing cycles. In a plane graph, a cycle C is said to be dividing if each closed region of the plane bounded by C contains infinitely many vertices.

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A classical theorem by Tutte assures the existence of a Hamilton cycle in every finite 4-connected planar graph. Extensions of this result to infinite graphs require a suitable concept of an infinite cycle. Such a concept was provided by Diestel and Kühn, who defined circles to be homeomorphic images of the unit circle in the Freudenthal compactification of the (locally finite) graph. With this definition we prove a partial extension of Tutte’s result to locally finite graphs. 1