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A generalization of dirac’s theorem

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Abstract

LetG be an (r+2)-connected graph in which every vertex has degree at leastd and which has at least 2d-r vertices. Then, for any pathQ of lengthr and vertexy not onQ, there is a cycle of length at least 2d-r containing bothQ andy.

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References

  1. J. A. Bondy,Personal Communication (1980).

  2. J. A. Bondy, Irith Ben-Arroyo Hartman andS. C. Locke, A new proof of a theorem of Dirac.Congressus Numerantium 32 (Proceedings of the Twelfth Southeastern Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, Louisiana, 1981), 131–136.

  3. J. A. Bondy andU. S. R. Murty,Graph Theory with Applications. Elsevier North Holland, New York (1976).

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  5. H. Enomoto, Long paths and large cycles in finite graphs.J. Graph Theory 8 (1984), 287–301.

    MATH  MathSciNet  Google Scholar 

  6. P. Erdős andT. Gallai, On maximal paths and circuits in graphs.Acta Math. Sci. Hung. 10 (1959), 337–356.

    Article  Google Scholar 

  7. M. Grötschel, Graphs with cycles containing given paths.Ann. Discrete Math. 1 (1977), 233–245.

    Article  Google Scholar 

  8. Irith Ben-Arroyo Hartman, Long cycles generate the cycle space of a graph.Europ. J. Combinatorics 4 (1983), 237–246.

    MATH  MathSciNet  Google Scholar 

  9. S. C. Locke, Some Extremal Properties of Paths, Cycles andk-colourable Subgraphs of Graphs.Ph. D. Thesis, University of Waterloo (1982).

  10. S. C. Locke, A basis for the cycle space of a 3-connected graph.Annals of Discrete Math., to appear.

  11. S. C. Locke, A basis for the cycle space of a 2-connected graph.Europ. J. Combinatorics, to appear.

  12. L. Lovász,Combinatorial Problems and Exercises. North-Holland (1979).

  13. K. Menger, Zur allgemeinen Kurventheorie.Fund. Math. 10 (1927), 96–115.

    MATH  Google Scholar 

  14. H.-J. Voss, Bridges of longest circuits and of longest paths in graphs.Beitrage zur Graphentheorie und deren Anwendungen (Proceedings of the International Colloquium, Oberhof, 1977), 275–286.

  15. H.-J. Voss andC. Zuluaga, Maximal gerade und ungerade Kreise in Graphen, I.Wiss. Z. Tech. Hochsch. Ilmenau 23 (1977), 57–70.

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Florida Atlantic University, 33431, Boca Raton, Fla, USA

    Stephen C. Locke

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  1. Stephen C. Locke
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Locke, S.C. A generalization of dirac’s theorem. Combinatorica 5, 149–159 (1985). https://doi.org/10.1007/BF02579378

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  • DOI: https://doi.org/10.1007/BF02579378

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