, Volume 39, Issue 1, pp 135–151 | Cite as

Long Cycles in Locally Expanding Graphs, with Applications

  • Michael KrivelevichEmail author


We provide sufficient conditions for the existence of long cycles in locally expanding graphs, and present applications of our conditions and techniques to Ramsey theory, random graphs and positional games.

Mathematics Subject Classification (2000)

05C38 05C35 05D10 05C80 05C57 


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  1. [1]
    M. Ajtai, J. Komlós and E. Szemerédi: The longest path in a random graph, Combinatorica 1 (1981), 1–12.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. Alon, M. Krivelevich and P. Seymour: Long cycles in critical graphs, J. Graph Th. 35 (2000), 193–196.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. Balogh, B. Csaba, M. Pei and W. Samotij: Large bounded degree trees in expanding graphs, Electr. J. Combin. 17 (2010), Publ. R6.MathSciNetzbMATHGoogle Scholar
  4. [4]
    J. Beck: On size Ramsey number of paths, trees, and circuits. I, J. Graph Th. 7 (1983), 115–129.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    J. Beck: Combinatorial games: Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications 114, Cambridge U. Press, Cambridge, 2008.CrossRefGoogle Scholar
  6. [6]
    M. Bednarska-Bzdȩga: On weight function methods in Chooser-Picker games, Theor. Comp. Sci. 475 (2013), 21–33.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Bednarska-Bzdȩga, D. Hefetz, M. Krivelevich and T. Łuczak: Manipulative waiters with probabilistic intuition, Combin. Probab. Comput. 25 (2016), 823–849.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Bednarska and T. Łuczak: Biased positional games and the phase transition, Random Struct. Alg. 18 (2001), 141–152.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Bednarska and O. Pikhurko: Biased positional games on matroids, Eur. J. Combin. 26 (2005), 271–285.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Bednarska and O. Pikhurko: Odd and even cycles in Maker-Breaker games, Eur. J. Combin. 29 (2008), 742–745.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    I. Ben-Eliezer, M. Krivelevich and B. Sudakov: The size Ramsey number of a directed path, J. Combin. Th. Ser. B 102 (2012), 743–755.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    I. Ben-Eliezer, M. Krivelevich and B. Sudakov: Long cycles in subgraphs of (pseudo)random directed graphs, J. Graph Th. 70 (2012), 284–296.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. Bierbrauer and A. Gyárfás: On (n;k)-colorings of complete graphs, Congressus Num. 58 (1987), 123–139.MathSciNetzbMATHGoogle Scholar
  14. [14]
    T. Bohman, A. Frieze, M. Krivelevich, P.-S. Loh and B. Sudakov: Ramsey games with giants, Random Struct. Alg. 38 (2011), 1–32.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    S. Brandt, H. Broersma, R. Diestel and M. Kriesell: Global connectivity and expansion: long cycles in f-connected graphs, Combinatorica 26 (2006), 17–36.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    J. Cain, P. Sanders and N. Wormald: The random graph threshold for korientability and a fast algorithm for optimal multiple-choice allocation, Proc 18th ACM-SIAM Symp. Discr. Alg. (SODA’07), 2007.Google Scholar
  17. [17]
    O. Dean and M. Krivelevich: Client Waiter games on complete and random graphs, Electron. J. Combin. 23 (4) (2016), P4.38.MathSciNetzbMATHGoogle Scholar
  18. [18]
    P. Dembowski: Finite Geometries, Springer Verlag, Berlin, 1968.CrossRefzbMATHGoogle Scholar
  19. [19]
    A. Dudek and P. Prałat: An alternative proof of the linearity of the size-Ramsey number of paths, Combin. Probab. Comput. 24 (2015), 551–555.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    A. Dudek and P. Prałat: On some multicolour Ramsey properties of random graphs, SIAM J. Discr. Math. 31 (2017), 2079–2092.CrossRefzbMATHGoogle Scholar
  21. [21]
    R. J. Faudree and R. H. Schelp: A survey of results on the size Ramsey number, in: Paul Erdos and his mathematics II, Bolyai Soc. Math. Stud. Vol. 11, 2002, 291–309.zbMATHGoogle Scholar
  22. [22]
    D. Fernholz and V. Ramachandran: The k-orientability Thresholds for Gn;p, Proc 18th ACM-SIAM Symp. Discr. Alg. (SODA’07), 2007.Google Scholar
  23. [23]
    J. Friedman and N. Pippenger: Expanding graphs contain all small trees, Combinatorica 7 (1987), 71–76.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Z. Füredi: Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1 (1981), 155–162.MathSciNetCrossRefGoogle Scholar
  25. [25]
    A. Gyárfás: Partition coverings and blocking sets in hypergraphs (in Hungarian), Commun. Comput. Automat. Inst. Hungar. Acad. Sci. 71 (1977).Google Scholar
  26. [26]
    P. Haxell: Tree embeddings, J. Graph Th. 36 (2001), 121–130.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    P. E. Haxell, Y. Kohayakawa and T. Łuczak: The induced size-Ramsey number of cycles, Combin. Probab. Comput. 4 (1995), 217–239.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    D. Hefetz, M. Krivelevich, M. Stojaković and T. Szabó: Positional Games, Birkhäuser, 2014.CrossRefzbMATHGoogle Scholar
  29. [29]
    D. Hefetz, M. Krivelevich and W. E. Tan: Waiter-Client and Client-Waiter planarity, colorability and minor games, Discr. Math. 339 (2016), 1525–1536.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    D. Hefetz, M. Krivelevich and W. E. Tan: Waiter-Client and Client-Waiter Hamiltonicity games on random graphs, Eur. J. Combin. 63 (2017), 26–43.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    S. Hoory, N. Linial and A. Wigderson: Expander graphs and their applications, Bull. Amer. Math. Soc. (N.S.) 43 (2006), 439–561.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    M. Krivelevich, C. Lee and B. Sudakov: Long paths and cycles in random subgraphs of graphs with large minimum degree, Random Struct. Alg. 46 (2015), 320–345.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    M. Krivelevich, K. Panagiotou, M. Penrose and C. McDiarmid: Random graphs, Geometry and Asymptotic Structure, Cambridge U. Press, Cambridge, 2016.CrossRefGoogle Scholar
  34. [34]
    M. Krivelevich and B. Sudakov: Pseudo-random graphs. in: More sets, graphs and numbers, E. Györi, G. O. H. Katona and L. Lovász, Eds., Bolyai Soc. Math. Stud. Vol. 15, 2006, 199–262.CrossRefzbMATHGoogle Scholar
  35. [35]
    M. Krivelevich and B. Sudakov: The phase transition in random graphs a simple proof, Random Struct. Alg. 43 (2013), 131–138.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    S. Letzter: Path Ramsey number for random graphs, Combin. Probab. Comput. 25 (2016), 612–622.MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    C. McDiarmid: On the method of bounded differences, in: Surveys in Combinatorics 1989, J. Siemons, Ed., London Math. Soc. Lect. Note Ser. 141, Cambridge U. Press, Cambridge, 1989.Google Scholar
  38. [38]
    A. Pokrovskiy: Partitioning edge-coloured complete graphs into monochromatic cycles and paths, J. Combin. Th. Ser. B 106 (2014), 70–97.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    L. Pósa: Hamiltonian circuits in random graphs, Discr. Math. 14 (1976), 359–364.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    O. Riordan: Long cycles in random subgraphs of graphs with large minimum degree, Random Struct. Alg. 45 (2014), 764–767.MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    R. Spöhel, A. Steger and H. Thomas: Coloring the edges of a random graph without a monochromatic giant component, Electron. J. Combin. 17 (2010), no. 1, Research Paper 133.MathSciNetzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018

Authors and Affiliations

  1. 1.School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael

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