On end degrees and infinite cycles in locally finite graphs

Abstract

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.

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References

  1. [1]

    Th. Andreae: Über maximale Systeme von kantendisjunkten unendlichen Wegen in Graphen, Math. Nachr. 101 (1981), 219–228.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    R. Diestel: The cycle space of an infinite graph, Comb., Probab. Comput. 14 (2005), 59–79.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    R. Diestel: Graph Theory (3rd ed.), Springer-Verlag, 2005.

  4. [4]

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

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

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

    Article  MathSciNet  Google Scholar 

  6. [6]

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

    MATH  Article  Google Scholar 

  7. [7]

    G. A. Dirac: Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 69–81.

    Article  MathSciNet  Google Scholar 

  8. [8]

    R. Halin: A note on Menger’s theorem for infinite locally finite graphs, Abh. Math. Sem. Univ. Hamburg 40 (1974), 111–114.

    MATH  MathSciNet  Article  Google Scholar 

  9. [9]

    D. W. Hall and G. L. Spencer: Elementary Topology, John Wiley, New York, 1955.

    Google Scholar 

  10. [10]

    F. Laviolette: Decompositions of infinite graphs: Part II — Circuit decompositions, J. Comb. Theory, Ser. B 94(2) (2005), 278–333.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    W. Mader: Homomorphieeigenschaften und mittlere Kantendichte von Graphen, Math. Ann. 174 (1967), 265–268.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    C. St. J. A. Nash-Williams: Decomposition of graphs into two-way infinite paths, Can. J. Math. 15 (1963), 479–485.

    MATH  MathSciNet  Google Scholar 

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Correspondence to Henning Bruhn.

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Bruhn, H., Stein, M. On end degrees and infinite cycles in locally finite graphs. Combinatorica 27, 269 (2007). https://doi.org/10.1007/s00493-007-2149-0

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Mathematics Subject Classification (2000)

  • 05C38
  • 05C45