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The minimum degree and connectivity of a graph

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Theory and Applications of Graphs

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 642))

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Abstract

Relationships between the minimum degree of a connected graph and its connectivity (or edge-connectivity) are discussed. We give a sufficient condition for a connected graph G to contain at least two nonadjacent vertices of degree κ(G), where κ(G) denotes the connectivity of G.

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References

  1. M. Behzad and G. Chartrand, Introduction to the Theory of Graphs, Allyn and Bacon, Boston 1971.

    MATH  Google Scholar 

  2. C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam and London, American Elsevier, New York 1973.

    MATH  Google Scholar 

  3. G. Chartrand, A graph-theoretic approach to a communications problem, SIAM J. Appl. Math. 14 (1966), 778–781.

    Article  MathSciNet  MATH  Google Scholar 

  4. G. Chartrand and F. Harary, Graphs with prescribed connectivities, Theory of Graphs (P. Erdös and G. Katona, eds.), Akadémiai Kiadó, Budapest 1968, 61–63.

    Google Scholar 

  5. G. Chartrand, A. Kaugars, and D. R. Lick, Critically n-connected graphs, Proc. Amer. Math. Soc., 32 (1972), 63–68.

    MathSciNet  MATH  Google Scholar 

  6. G. A. Dirac, Minimally 2-connected graphs, J. Reine Angew. Math., 228 (1967), 204–216.

    MathSciNet  MATH  Google Scholar 

  7. D. P. Geller, Minimally strong digraphs, Proc. Edinburgh Math. Soc., 17 (Series II) (1970), 15–22.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Halin, A theorem on n-connected graphs, J. Combinatorial Theory, 7 (1969), 150–154.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Halin, Studies on minimally n-connected graphs, Combinatorial Mathematics and Its Applications (D. J. A. Welsh, ed.), Academic Press, London and New York 1971, 129–136.

    Google Scholar 

  10. F. Harary, Graph Theory, Addison-Wesley, Reading, Massachusetts 1969).

    Google Scholar 

  11. T. Kameda, Note on Halin's theorem on minimally connected graphs, J. Combinatorial Theory, 17 (Series B) (1974), 1–4.

    MathSciNet  MATH  Google Scholar 

  12. A. Kaugars, A Theorem on the Removal of Vertices from Blocks, Senior Thesis, Kalamazoo College, Kalamazoo (Michigan), 1968.

    Google Scholar 

  13. L. Lesniak, Results on the edge-connectivity of graphs, Discrete Mathematics, 8 (1974), 351–354.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. R. Lick, Critically and minimally n-connected graphs, The Many Facets of Graph Theory (G. Chartrand and S. F. Kapoor, eds.), Lecture Notes in Mathematics 110, Springer-Verlag, Berlin, Heidelberg, New York 1969, 199–205.

    Chapter  Google Scholar 

  15. D. R. Lick, Minimally n-line connected graphs, J. Reine Angew. Math., 252 (1972), 178–182.

    MathSciNet  MATH  Google Scholar 

  16. W. Mader, Eine Eigenschaft der Atome endlicher Graphen, Archiv der Mathematik, 22 (1971), 333–336.

    Article  MathSciNet  Google Scholar 

  17. W. Mader, Minimale n-fach kantenzusammenhängende Graphen, Math. Ann., 191 (1971), 21–28.

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Mader, Ecken vam Grad n in minimalen n-fach zusammenhängenden Graphen, Archiv Math., 23(1972), 219–224.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Mader, Ecken vom Innen-und Außengrad n in minimal n-fach kantenzusammenhängenden Digraphen, Archiv Math., 25(1974), 107–112.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Nebeský, A theorem on 2-connected graphs, Casopis pest. mat., 100(1975), 116–117.

    MathSciNet  MATH  Google Scholar 

  21. L. Nebeský, On the minimum degree and edge-connectivity of a graph, to appear.

    Google Scholar 

  22. L. Nebeský, An upper bound for the minimum degree of a graph, submitted for publication.

    Google Scholar 

  23. L. Nebeský, On induced subgraphs of a block, submitted for publication.

    Google Scholar 

  24. M. D. Plummer, On minimal blocks, Trans. Amer. Math. Soc. 134 (1968), 85–94.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Charles University, Prague, Czechoslovakia

    Ladislav Nebeský

Authors
  1. Ladislav Nebeský
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Yousef Alavi  & Don R. Lick  & 

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© 1978 Springer-Verlag Berlin Heidelberg

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Nebeský, L. (1978). The minimum degree and connectivity of a graph. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070398

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  • DOI: https://doi.org/10.1007/BFb0070398

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08666-6

  • Online ISBN: 978-3-540-35912-8

  • eBook Packages: Springer Book Archive

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