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Contractible and Removable Edges in 3-Connected Infinite Graphs

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Abstract

Several results concerning contractible and removable edges in 3-connected finite graphs are extended to infinite graphs. First, we prove that every 3-connected locally finite infinite graph has infinitely many removable edges. Next, we prove that for any 3-connected graph \(G\), if \(x\) is a finite degree vertex in \(G\) and is not incident to any contractible edges, then \(G-x\) is a finite cycle or contains a border pair. As a result, every 3-connected locally finite infinite graph contains infinitely many contractible edges. Lastly, it is shown that for any 3-connected locally finite infinite graph \(G\) which is triangle-free or has minimum degree at least 4, the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of \(G\) is topologically 2-connected.

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References

  1. Ando, K., Enomoto, H., Saito, A.: Contractible edges in 3-connected graphs. J. Comb. Theory Ser. B 42, 87–93 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barnette, D.W., Grünbaum, B.: On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs. Many facets of graph theory. Lecture Notes in Mathematics, vol. 110, pp. 27–40. Springer, New York (1966)

  3. Chan, T.: Contractible edges in 2-connected locally finite graphs, submitted.

  4. Dean, N.: Distribution of contractible edges in k-connected graphs. J. Comb. Theory Ser. B 48, 1–5 (1990)

    Article  MATH  Google Scholar 

  5. Diestel, R.: Graph theory, 3rd edn. Springer, New York (2005)

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  9. Holton, D.A., Jackson, B., Saito, A., Wormald, N.C.: Removable edges in 3-connected graphs. J. Graph Theory 14, 465–473 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kriesell, M.: Contractible subgraphs in 3-connected graphs. J. Comb. Theory Ser. B 80, 32–48 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kriesell, M.: A survey on contractible edges in graphs of a prescribed vertex connectivity. Graphs Comb. 18, 1–30 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kriesell, M.: Average degree and contractibility. J. Graph Theory 51, 205–224 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  13. Mader, W.: Generalizations of critical connectivity of graphs. Discret. Math. 72, 267–283 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  14. Su, J.: The number of removable edges in 3-connected graphs. J. Comb. Theory Ser. B 75, 74–87 (1999)

    Article  MATH  Google Scholar 

  15. Tutte, W.T.: A theory of 3-connected graphs. Nederl. Akad. Wet. Proc. Ser A 64, 441–455 (1961)

    MATH  MathSciNet  Google Scholar 

  16. Wu, H.: Contractible elements in graphs and matroids. Comb. Probab. Comput. 12, 457–465 (2003)

    Article  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the referees for pointing out errors and giving helpful suggestions.

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  1. Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, 20146 , Hamburg, Germany

    Tsz Lung Chan

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  1. Tsz Lung Chan
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Correspondence to Tsz Lung Chan.

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Chan, T.L. Contractible and Removable Edges in 3-Connected Infinite Graphs. Graphs and Combinatorics 31, 871–883 (2015). https://doi.org/10.1007/s00373-014-1431-3

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  • DOI: https://doi.org/10.1007/s00373-014-1431-3

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