The Cycle Space of an Infinite Graph

  • Authors:

  • Reinhard Diestel

  • Affiliations:

  • Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany (e-mail: diestel@math.uni-hamburg.de)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2005

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Abstract

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new ‘singular’ approach 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.Our approach permits the extension to infinite graphs of standard results about finite graph homology – such as cycle–cocycle duality and Whitney's theorem, Tutte's generating theorem, MacLane's planarity criterion, the Tutte/Nash-Williams tree packing theorem – whose infinite versions would otherwise fail. A notion of end degrees motivated by these results opens up new possibilities for an ‘extremal’ branch of infinite graph theory.Numerous open problems are suggested.