# On end degrees and infinite cycles in locally finite graphs

```@article{Bruhn2007OnED,
title={On end degrees and infinite cycles in locally finite graphs},
author={Henning Bruhn and Maya Jakobine Stein},
journal={Combinatorica},
year={2007},
volume={27},
pages={269-291}
}```
• Published 1 May 2007
• Mathematics
• Combinatorica
We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in 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.
39 Citations
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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
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Comb.
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The spanning trees whose fundamental cycles generate this cycle space are characterized, and infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decomposition into single cycles and orthogonality to cuts are proved.
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• Mathematics
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A new ‘singular’ approach is presented that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of \$S^1\$ in the space formed by the graph together with its ends.
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