On Infinite Cycles I

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 S 1 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.

This is a preview of subscription content, access via your institution.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Reinhard Diestel.

Additional information

To the memory of C. St. J. A. Nash-Williams

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Diestel, R., Kühn, D. On Infinite Cycles I. Combinatorica 24, 69–89 (2004). https://doi.org/10.1007/s00493-004-0005-z

Download citation

Mathematics Subject Classification (2000):

  • 5C10
  • 5C38
  • 5C45
  • 57M15