Cycle lengths in sparse graphs

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References

  1. [1]

    N. Alon, M. Krivelevich and B. Sudakov: Turán numbers of bipartite graphs and related Ramsey-type questions, Combinatorics, Probability and Computing 12 (2003), 477–494.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    N. Alon, L. Rónyai and T. Szabó: Norm-graphs: variations and applications, J. Combinatorial Theory Ser. B 76 (1999), 280–290.

    MATH  Article  Google Scholar 

  3. [3]

    B. Bollobás: Cycles modulo k, Bull. London Math. Soc. 9 (1977), 97–98.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    B. Bollobás and A. Thomason: Weakly pancyclic graphs, J. Combinatorial Theory Ser. B 77 (1999), 121–137.

    MATH  Article  Google Scholar 

  5. [5]

    A. Bondy: Pancyclic graphs I, J. Combinatorial Theory Ser. B 11 (1971), 80–84.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    A. Bondy and M. Simonovits: Cycles of even length in graphs, J. Combinatorial Theory Ser. B 16 (1974), 97–105.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    M. N. Ellingham and D. K. Menser: Girth, minimum degree, and circumference; J. Graph Theory 34(3) (2000), 221–233.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    P. Erdős: Some of my favourite problems in various branches of combinatorics, Matematiche (Catania) 47 (1992), 231–240.

    MathSciNet  Google Scholar 

  9. [9]

    P. Erdős: Some of my favorite solved and unsolved problems in graph theory, Quaestiones Math. 16 (1993), 333–350.

    MathSciNet  Google Scholar 

  10. [10]

    P. Erdős: Some old and new problems in various branches of combinatorics, in: Graphs and combinatorics, (Marseille, 1995); also: Discrete Mathematics 165–166(15) (1997), 227–231.

  11. [11]

    P. Erdős, R. Faudree, C. Rousseau and R. Schelp: The number of cycle lengths in graphs of given minimum degree and girth, Discrete Mathematics 200 (1999), 55–60.

    Article  MathSciNet  Google Scholar 

  12. [12]

    G. Fan: Distribution of cycle lengths in graphs, J. Combinatorial Theory Ser. B 84 (2002), 187–202.

    MATH  Article  Google Scholar 

  13. [13]

    R. Gould, P. Haxell and A. Scott: A note on cycle lengths in graphs, Graphs and Combinatorics 18 (2002), 491–498.

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

    A. Gyárfás: Graphs with κ odd cycle lengths, Discrete Mathematics 103 (1992), 41–48.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    A. Gyárfás, J. Komlós and E. Szemerédi: On the distribution of cycle lengths in graphs, J. Graph Theory 8 (1984), 441–462.

    MATH  Article  MathSciNet  Google Scholar 

  16. [16]

    J. Kollár, L. Rónyai and T. Szabó: Norm-graphs and bipartite Turán numbers, Combinatorica 16(3) (1996), 399–406.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    T. Kővári, V. T. Sós and P. Turán: On a problem of K. Zarankiewicz, Colloquium Math. 3 (1954), 50–57.

    MathSciNet  Google Scholar 

  18. [18]

    L. Lovász: Combinatorial Problems and Exercises, 2nd Ed., North-Holland, Amsterdam, 1993.

    Google Scholar 

  19. [19]

    A. Lubotzky, R. Phillips and P. Sarnak: Ramanujan Graphs, Combinatorica 8(3) (1988), 261–277.

    MATH  Article  MathSciNet  Google Scholar 

  20. [20]

    G. Margulis: Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators, Problems Inform. Transmission 24 (1988), 39–46.

    MATH  MathSciNet  Google Scholar 

  21. [21]

    O. Ore: On a graph theorem by Dirac, J. Combinatorial Theory 2 (1967), 383–392.

    MATH  Article  MathSciNet  Google Scholar 

  22. [22]

    L. Pósa: Hamiltonian circuits in random graphs, Discrete Mathematics 14 (1976), 359–364.

    MATH  Article  MathSciNet  Google Scholar 

  23. [23]

    J. Verstraëte: On arithmetic progressions of cycle lengths in graphs, Combinatorics Probability and Computing 9 (2000), 369–373.

    MATH  Article  MathSciNet  Google Scholar 

  24. [24]

    C. Q. Zhang: Circumference and girth, J. Graph Theory 13(4) (1989), 485–490.

    Article  MathSciNet  Google Scholar 

  25. [25]

    B. Z. Zhao: The circumference and girth of a simple graph (Chinese), J. Northeast Univ. Tech. 13(3) (1992), 294–296.

    MathSciNet  Google Scholar 

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Correspondence to Benny Sudakov.

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Research supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant, Alfred P. Sloan fellowship, and the State of New Jersey.

Research supported by an Alfred P. Sloan Fellowship.

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Sudakov, B., Verstraëte, J. Cycle lengths in sparse graphs. Combinatorica 28, 357–372 (2008). https://doi.org/10.1007/s00493-008-2300-6

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C38