Cycles and Paths in Triangle-Free Graphs

  • Stephan Brandt
Part of the Algorithms and Combinatorics book series (AC, volume 14)


Let G bea triangle-free graph of order n and minimum degree δ > n/3. We will determine all lengths of cycles occurring in G. In particular, the length of a longest cycle or path in G is exactly the value admitted by the independence number of G. This value can be computed in time O(n 2.5) using the matching algorithm of Micali and Vazirani. An easy consequence is the observation that triangle-free non-bipartite graphs with \(\delta \geqslant \frac{3} {8}n\) are hamiltonian.


Bipartite Graph Perfect Match Minimum Degree Hamiltonian Cycle Maximum Match 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Alon, Tough Ramsey graphs without short cycles, Manuscript (1993).Google Scholar
  2. 2.
    D. Amar, A condition for a hamiltonian graph to be bipancyclic, Discr. Math. 102(1991), 221–227.MathSciNetCrossRefGoogle Scholar
  3. 3.
    D. Amar, I. Fournier and A. Germa, Pancyclism in Chvátal—Erdős graphs, Graphs and Combinatorics 7 (1991), 101–112.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    D. Amar and Y. Manoussakis, Cycles and paths of many lengths in bipartite digraphs, J. Combin. Theory Ser. B 50 (1990), 254–264.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    D. Bauer, J. van den Heuvel and E. Schmeichel, Toughness and triangle-free graphs, Preprint (1993).Google Scholar
  6. 6.
    C. Berge, Sur le couplage maximum d’un graphe, C. R. Acad. Sci. Paris (A) 247 (1958), 258–259.MathSciNetzbMATHGoogle Scholar
  7. 7.
    S. Brandt, Sufficient conditions for graphs to contain all subgraphs of a given type, Ph.D. thesis, Freie Universität Berlin, 1994.Google Scholar
  8. 8.
    S. Brandt, R. J. Faudree and W. Goddard, Weakly pancyclic graphs, in preparation.Google Scholar
  9. 9.
    G. Chartrand AND L. Lesniak, Graphs & Digraphs (2nd edition), Wadsworth & Brooks/Cole, Pacific Grove, 1986.zbMATHGoogle Scholar
  10. 10.
    V. Chvátal, Tough graphs and hamiltonian cycles, Discr. Math. 5 (1973), 215–228.zbMATHCrossRefGoogle Scholar
  11. 11.
    V. Chvátal and P. Erdős, A note on hamiltonian circuits, Discr. Math. 2 (1972), 111–113.zbMATHCrossRefGoogle Scholar
  12. 12.
    G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. (3) 2 (1952), 69–81.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    H. Enomoto, J. van den Heuvel, A. Kaneko, and A. Saito, Relative length of long paths and cycles in graphs with large degree sums, Preprint (1993).Google Scholar
  14. 14.
    R. Häggkvist, On the structure of non-hamiltonian graphs, Combin. Prob, and Comp. 1 (1992), 27–34.zbMATHCrossRefGoogle Scholar
  15. 15.
    Dingjun Lou, The Chvatal-Erdős condition for cycles in triangle-free graphs, to appear.Google Scholar
  16. 16.
    S. Micali and V. V. Vazirani, An O(V 1/2 E) algorithm for finding maximum matching in general graphs, Proc. 21st Ann. Symp. on Foundations of Computer Sc. IEEE, New York (1980), 17–27.Google Scholar
  17. 17.
    O. Ore, Note on Hamiltonian circuits, Amer. Math. Monthly 67 (1960), 55.MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    W. T. Tutte, The factorization of linear graphs, J. London Math. Soc. 22 (1947), 107–111.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    H. J. Veldman, Personal communication, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Stephan Brandt
    • 1
  1. 1.Fachbereich MathematikFreie Universität BerlinBerlinGermany

Personalised recommendations