Cycle Lengths in Expanding Graphs


For a positive constant α a graph G on n vertices is called an α-expander if every vertex set U of size at most n/2 has an external neighborhood whose size is at least α|U|. We study cycle lengths in expanding graphs. We first prove that cycle lengths in α-expanders are well distributed. Specifically, we show that for every 0 < α ≤ 1 there exist positive constants n0, C and A = O(1/α) such that for every α-expander G on nn0 vertices and every integer \(\ell \in \left[ {C\log n,\tfrac{n}{C}} \right]\), G contains a cycle whose length is between \(\ell\) and \(\ell\)+A; the order of dependence of the additive error term A on α is optimal. Secondly, we show that every α-expander on n vertices contains \(\Omega \left( {\tfrac{{{\alpha ^3}}}{{\log (1/\alpha )}}} \right)\) different cycle lengths. Finally, we introduce another expansion-type property, guaranteeing the existence of a linearly long interval in the set of cycle lengths. For β > 0 a graph G on n vertices is called a β-graph if every pair of disjoint sets of size at least βn are connected by an edge. We prove that for every \(\beta < 1/20\) there exist positive constants \({b_1} = O\left( {\tfrac{1}{{\log (1 - \beta )}}} \right)\) and b2 = O(β) such that every β-graph G on n vertices contains a cycle of length \(\ell\) for every integer \(\ell \in \left[ {{b_1}\log n,\left( {1 - {b_2}} \right)n} \right]\); the order of dependence of b1 and b2 on β is optimal.

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  1. [1]

    N. Alon: Eigenvalues and expanders, Combinatorial 6 (1986), 83–96.

    MathSciNet  Article  Google Scholar 

  2. [2]

    N. ALon and N. Linial: Cycles of length 0 modulo k in directed graphs, Journal of Combinatorial Theory, Ser. B 47 (1989), 114–119.

    MathSciNet  Article  Google Scholar 

  3. [3]

    N. Alon and J. H. Spencer: The Probabilistic Method, 4th edition, Wiley, New York, 2015.

    Google Scholar 

  4. [4]

    J. Balogh, B. CSABA, M. Pei and W. Samotij: Large bounded degree trees in expanding graphs, Electronic Journal of Combinatorics 17, Research Paper 6, 2010.

  5. [5]

    B. Bollobás: Cycles modulo k, Bulletin of the London Mathematical Society 9 (1977), 97–98.

    MathSciNet  Article  Google Scholar 

  6. [6]

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

    MathSciNet  Article  Google Scholar 

  7. [7]

    G. T. Chen and A. Saito: Graphs with a Cycle of Length Divisible by Three, Journal of Combinatorial Theory, Ser. B 60 (1994), 277–292.

    MathSciNet  Article  Google Scholar 

  8. [8]

    N. Dean, L. Lesniak and A. Saito: Cycles of length 0 modulo 4 in graphs, Discrete Mathematics 121 (1993), 37–49.

    MathSciNet  Article  Google Scholar 

  9. [9]

    P. Erdős: Some recent problems and results in graph theory, combinatorics and number theory, in: Proceedings of the 7th Southeast Conference on Combinatorics, Graph Theory and Computing, 3-14, 1976.

    Google Scholar 

  10. [10]

    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.

    MathSciNet  Article  Google Scholar 

  11. [11]

    G. Fan: Distribution of Cycle Lengths in Graphs, Journal of Combinatorial Theory, Ser. B 82 (2002), 187–202.

    MathSciNet  Article  Google Scholar 

  12. [12]

    J. Friedman and N. Pippenger: Expanding graphs contain all small trees, Combi-natorica 7 (1987), 71–76.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    J. Gao, Q. Huo, C. H. Liu and J. Ma: A unified proof of conjectures on cycle lengths in graphs, arXiv:1904.08126 [math.CO], 2019.

    Google Scholar 

  14. [14]

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

    MathSciNet  Article  Google Scholar 

  15. [15]

    P. E. Haxell: Tree embeddings, Journal of Graph Theory 36 (2001), 121–130.

    MathSciNet  Article  Google Scholar 

  16. [16]

    D. Hefetz, M. Krivelevich and T. Szabó: Hamilton cycles in highly connected and expanding graphs, Combinatorica 29 (2009), 547–568.

    MathSciNet  Article  Google Scholar 

  17. [17]

    S. Hoory, N. Linial and A. Wigderson: Expander graphs and their applications, Bulletin of the American Mathematical Society 43 (2006), 439–561.

    MathSciNet  Article  Google Scholar 

  18. [18]

    A. Kostochka, B. Sudakov and J. Verstraëte: Cycles in triangle-free graphs of large chromatic number, Combinatorica 37 (2017), 481–494.

    MathSciNet  Article  Google Scholar 

  19. [19]

    M. Krivelevich: Long paths and Hamiltonicity in random graphs, Random Graphs, Geometry and Asymptotic Structure, London Mathematical Society Student Texts 84, Cambridge University Press, 4–27, 2016.

    Google Scholar 

  20. [20]

    M. Krivelevich: Finding and using expanders in locally sparse graphs, SIAM Journal on Discrete Mathematics 32 (2018), 611–623.

    MathSciNet  Article  Google Scholar 

  21. [21]

    M. Krivelevich: Expanders - how to find them, and what to find in them, in: Surveys in Combinatorics 2019, A. Lo et al., Eds., London Mathematical Society Lecture Notes 456, 115–142, 2019.

    Google Scholar 

  22. [22]

    C. H. Liu and J. Ma: Cycle lengths and minimum degree of graphs, Journal of Combinatorial Theory, Ser. B 128 (2018), 66–95.

    MathSciNet  Article  Google Scholar 

  23. [23]

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

    MathSciNet  Article  Google Scholar 

  24. [24]

    P. Mihók and I. Schiermeyer: Cycle lengths and chromatic number of graphs, Discrete Mathematics 286 (2004), 147–149.

    MathSciNet  Article  Google Scholar 

  25. [25]

    B. Sudakov and J. Verstraëte: Cycle lengths in sparse graphs, Combinatorica 28 (2008), 357–372.

    MathSciNet  Article  Google Scholar 

  26. [26]

    B. Sudakov and J. Verstraëte: The Extremal Function for Cycles of Length l mod k, Electronic Journal of Combinatorics 24, Paper 1.7, 2016.

  27. [27]

    C. Thomassen: Graph decomposition with applications to subdivisions and path systems modulo k, Journal of Graph Theory 7 (1983), 261–271.

    MathSciNet  Article  Google Scholar 

  28. [28]

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

    MathSciNet  Article  Google Scholar 

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The authors are grateful to Rajko Nenadov for his contribution to this paper. The authors also wish to thank Asaf Cohen, Wojciech Samotij and Leonid Vishnevsky for their input and remarks.

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Correspondence to Michael Krivelevich.

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Research supported in part by USA-Israel BSF grant 2018267, and by ISF grant 1261/17.

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Friedman, L., Krivelevich, M. Cycle Lengths in Expanding Graphs. Combinatorica 41, 53–74 (2021).

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Mathematics Subject Classification (2020)

  • 05C38
  • 05C48
  • 05C35