The bond and cycle spaces of an infinite graph

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.