Eulerian edge sets in locally finite graphs

Abstract

In a finite graph, an edge set Z is an element of the cycle space if and only if every vertex has even degree in Z. We extend this basic result to the topological cycle space, which allows infinite circuits, of locally finite graphs. In order to do so, it becomes necessary to attribute a parity to the ends of the graph.

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References

  1. [1]

    H. Bruhn: The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits, J. Combin. Theory (Series B) 92 (2004), 235–256.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    H. Bruhn, R. Diestel and M. Stein: Cycle-cocycle partitions and faithful cycle covers for locally finite graphs, J. Graph Theory 50 (2005), 150–161.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    H. Bruhn and M. Stein: MacLane’s planarity criterion for locally finite graphs, J. Combin. Theory (Series B) 96 (2006), 225–239.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    H. Bruhn and M. Stein: On end degrees and infinite cycles in locally finite graphs, Combinatorica 27(3) (2007), 269–291.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    Q. Cui, J. Wang and X. Yu: Hamilton circles in infinite planar graphs, J. Combin. Theory (Series B) 99 (2009), 110–138.

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

    R. Diestel: Graph theory (3rd edition), Springer-Verlag, 2005.

  7. [7]

    R. Diestel and D. Kühn: On infinite cycles I, Combinatorica 24(1) (2004), 69–89.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    R. Diestel and D. Kühn: On infinite cycles II, Combinatorica 24(1) (2004), 91–116.

    MathSciNet  Article  Google Scholar 

  9. [9]

    R. Diestel and D. Kühn: Topological paths, cycles and spanning trees in infinite graphs; Europ. J. Combinatorics 25 (2004), 835–862.

    MATH  Article  Google Scholar 

  10. [10]

    A. Georgakopoulos: Infinite Hamilton cycles in squares of locally finite graphs, Adv. Math. 220(3) (2009), 670–705.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    C. Thomassen and A. Vella: Graph-like continua, augmenting arcs, and Menger’s theorem; Combinatorica 28(5) (2008), 595–623.

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    A. Vella and R. B. Richter: Cycle spaces of topological spaces, J. Graph Theory 59 (2008), 115–144.

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Eli Berger.

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Berger, E., Bruhn, H. Eulerian edge sets in locally finite graphs. Combinatorica 31, 21 (2011). https://doi.org/10.1007/s00493-011-2572-0

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Mathematics Subject Classification (2000)

  • 05C38
  • 05C45