Combinatorica

, Volume 36, Issue 5, pp 601–621 | Cite as

Orientations of infinite graphs with prescribed edge-connectivity

Article

Abstract

We prove a decomposition result for locally finite graphs which can be used to extend results on edge-connectivity from finite to infinite graphs. It implies that every 4k-edge-connected graph G contains an immersion of some finite 2k-edge-connected Eulerian graph containing any prescribed vertex set (while planar graphs show that G need not containa subdivision of a simple finite graph of large edge-connectivity). Also, every 8k-edge connected infinite graph has a k-arc-connected orientation, as conjectured in 1989.

Mathematics Subject Classification (2000)

05C40 05C20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Aharoni and C. Thomassen: Infinite, highly connected digraphs with no twoarc-disjoint spanning trees, J. Graph Theory 13 (1989), 71–74.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J. BarÁt and M. Kriesell: What is on his mind?, Discrete Mathematics 310 (2010), 2573–2583.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    J. A. Bondy and U. S. R. Murty: Graph Theory with Applications, The MacMillanPress Ltd. (1976).Google Scholar
  4. [4]
    R. Diestel: Graph Theory, Springer Verlag (1997) and 4th edition (2010).CrossRefMATHGoogle Scholar
  5. [5]
    G. A. Dirac and C. Thomassen: Graphs in which every finite path is contained ina circuit, Math. Ann. 203 (1973), 65–75.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    J. Edmonds: Minimum partition of a matroid into independent subsets, J. Res. Nat.Bur. Standards Sect. B 69B (1965), 67–72.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    R. Halin: A note on Menger's theorem for infinite locally finite graphs, Abh. Math.Sem. Univ. Hamburg 40 (1974), 111–114.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    F. Laviolette: Decompositions of infinite Graphs: I–bond-faithful decompositions, Journal of Combinatorial Theory, Series B 94 (2005), 259–277.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    L. LovÁsz: On some connectivity properties of eulerian graphs, Acta Math. Acad.Sci. Hung. 28 (1976), 129–138.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    W. Mader: A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978), 145–164.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    B. Mohar and C. Thomassen: Graphs on Surfaces, Johns Hopkins University Press(2001).MATHGoogle Scholar
  12. [12]
    C. St. J. A. Nash-Williams: On orientations, connectivity and odd-vertex-pairingsin finite graphs, Canad. J. Math. 12 (1960), 555–567.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    C. St. J. A. Nash-Williams: Edge-disjoint spanning trees of finite graphs, J. Lon-don Math. Soc. 36 (1961), 445–450.MathSciNetMATHGoogle Scholar
  14. [14]
    C. St. J. A. Nash-Williams: Infinite graphs-a survey, J. Combin. Theory 3 (1967),286–301.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    C. St. J. A. Nash-Williams: Unexplored and semi-explored territories in graphtheory, in: New Directions in the Theory of Graphs (Proc. Third Ann Arbor Conf.,Univ. Michigan, Ann Arbor, MI, 1971), Academic Press, New York (1973), 149–186.Google Scholar
  16. [16]
    H. E. Robbins: Questions, discussions, and notes: a theorem on graphs, with anapplication to a problem of traffic control, Amer. Math. Monthly 46 (1939), 281–283.CrossRefGoogle Scholar
  17. [17]
    C. Thomassen: 2-Linked Graphs, Europ. J. Combinatorics 1 (1980), 371–378.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    C. Thomassen: Infinite graphs, in: Further Selected Topics in Graph Theory (L.W. Beineke and R.J. Wilson, eds.), Academic Press, London (1983), 129–160.Google Scholar
  19. [19]
    C. Thomassen: Configurations in graphs of large minimum degree, connectivity, orchromatic number, in: Combinatorial Mathematics: Proceedings of the Third Interna-tional Conference, New York 1985, Ann. New York Acad. Sci., New York 555 (1989), 402–412.MathSciNetGoogle Scholar
  20. [20]
    C. Thomassen: The weak 3-ow conjecture and the weak circular ow conjecture,J. Combin. Theory Ser. B. 102 (2012), 521–529.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    W. T. Tutte: On the problem of decomposing a graph into n connected factors, J.London Math. Soc. 36 (1961), 221–230.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkLyngbyDenmark

Personalised recommendations