On end degrees and infinite cycles in locally finite graphs
We introduce a natural extension of the vertex degree to ends. For the cycle space \({\mathcal C}(G)\) as proposed by \textit{R. Diestel} and \textit{D. Kühn} [Combinatorica 24, No. 1, 69--89, 91--116 (2004; Zbl 1063.05076)], which allows for infinite cycles, we prove that the edge set of a locally finite graph \(G\) lies in \({\mathcal C}(G)\) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.
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