A Combinatorial Classic — Sparse Graphs with High Chromatic Number

  • Jaroslav Nešetřil
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 25)


It seems that combinatorics, and graph theory in particular, reached mathematical maturity relatively recently. Perhaps as a result of this there are not too many essential stories which have determined the course of the subject over a long period, enduring stories which appear again and again as a source of inspiration and motivate and challenge research.


Chromatic Number Sparse Graph Cover Graph Ramsey Number Large Girth 
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.


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  1. [1]
    M. Ajtai, J. Komlós, E. Szemerédi, A note on Ramsey numbers, J. Combi. Th. Series A, 29 (1980), 354–360.CrossRefzbMATHGoogle Scholar
  2. [2]
    M. Ajtai, J. Komlós, E. Szemerédi, An O(n log n) sorting network, Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 1983, 1–9.Google Scholar
  3. [3]
    N. Alon, Discrete Mathematics: methods and challenges, Proc. of the International Congress of Mathematicians (ICM), Beijing 2002, China, Higher Education Press (2003), 119–135.Google Scholar
  4. [4]
    N. Alon, V. Rödl, Sharp bounds for some multicolor Ramsey numbers, Combinatorica 25 (2005), 125–141.CrossRefzbMATHMathSciNetGoogle Scholar
  5. [5]
    N. Alon, J. H. Spencer, The probabilistic method Second ed. Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, 2000.CrossRefzbMATHGoogle Scholar
  6. [6]
    O. Angel, A. Kechris, R. Lyons, Random Orderings and Unique Ergodicity of Automorphism Groups, arXiv: 1208.2389 (2012).Google Scholar
  7. [7]
    L. Barto, M. Kozik, New conditions for Taylor varieties and CSP, In: Proceedings of LICS’10, IEEE, 2010, 100–109.Google Scholar
  8. [8]
    S. Baum, M. Stiebitz, Coloring Of Graphs Without Short Odd Paths Between Vertices Of The Same Color Class, (preprint, TU Ilmenau).Google Scholar
  9. [9]
    D. Bokal, G. Fijavž, M. Juvan, P. M. Kayll, B. Mohar, The circular chromatic number of a digraph, J. Graph Theory 46 (2004), no. 3, 227–240.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    B. Bollobás, Random Graphs, Academic Press, 1985.Google Scholar
  11. [11]
    B. Bollobás, Modern Graph Theory, Springer-Verlag, 1998.Google Scholar
  12. [12]
    B. Bollobás and N. Sauer, Uniquely colorable graphs with large girth, Can. J. Math. 28(1976), 1340–1344.CrossRefzbMATHGoogle Scholar
  13. [13]
    J. A. Bondy, U. S. R. Murty, Graph theory, Graduate Texts in Mathematics, 244, Springer, 2008.Google Scholar
  14. [14]
    G. Brightwell, On the complexity of diagram testing, Order 10,4 (1983), 297–303.CrossRefMathSciNetGoogle Scholar
  15. [15]
    A. A. Bulatov, P. Jeavons, A. A. Krokhin, Classifying the Complexity of Constraints Using Finite Algebras, SIAM J. Comput. 34(3), 2005, 720–742.CrossRefzbMATHMathSciNetGoogle Scholar
  16. [17]
    G. Cherlin, S. Shelah, N. Shi, Universal graphs with forbidden subgraphs and algebraic closure, Adv. in Applied Math. 22 (1999), 454–491.CrossRefzbMATHMathSciNetGoogle Scholar
  17. [18]
    G. Cherlin, N. Shi, Graphs omitting a finite set of cycles, J. of Graph Th. 21 (1997), 351–355.MathSciNetGoogle Scholar
  18. [19]
    B. Codenotti, P. Pudlák, J. Resta, Some structural properties of low rank matrices related to computational complexity, Theoretical Computer Sci. 235 (2000), 89–107.CrossRefzbMATHGoogle Scholar
  19. [20]
    B. Descartes, A three colour problem, Eureka 21, 1947.Google Scholar
  20. [21]
    R. Diestel, Graph theory, Graduate Texts in Mathematics, 173, Springer, Heidelberg, 2010.CrossRefGoogle Scholar
  21. [22]
    T. Emden-Weinert, S. Hongardy, B. Kreutzer, Uniquelly colorable graphs and hardness of colouring of graphs of large girth, Comb. Prob. Comp. 7,4 (1998), 375–386.CrossRefzbMATHGoogle Scholar
  22. [23]
    P. Erdős, Graph theory and probability, Canad. J. Math. 11 (1959), 34–38.CrossRefMathSciNetGoogle Scholar
  23. [24]
    P. Erdős, Problems and results in combinatorial analysis and graph theory, In: Proof Techniques in Graph Theory, Academic Press, 1969, 27–35.Google Scholar
  24. [25]
    P. Erdős, A. Hajnal, On chromatic number of graphs and set-systems, Acta Math. Acad. Sci. Hungar. 17 (1966), 61–99.CrossRefMathSciNetGoogle Scholar
  25. [26]
    P. Erdős, J. Nešetřil, V. Rödl On Pisier type problems and results (combinatorial applications to number theory), In: Mathematics of Ramsey Theory, Springer 1990, 214–231.Google Scholar
  26. [27]
    P. Erdős, C. A. Rogers, The construction of certain graphs, Canad. J. Math. 14 (1962), 702–707.CrossRefMathSciNetGoogle Scholar
  27. [28]
    P. Erdős, E. Specker, On a theorem in the theory of relations and a solution of a problem of Knaster, Colloq. Math. 8 (1961), 19–21.MathSciNetGoogle Scholar
  28. [29]
    T. Feder, M. Y. Vardi, The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory, SIAM J. Comput. 28, 1 (1999), 57–104.CrossRefMathSciNetGoogle Scholar
  29. [30]
    R. Graham and B. Rothschild and J. Spencer, Ramsey Theory, Wiley, New York, 1990.zbMATHGoogle Scholar
  30. [31]
    D. Greenwell and L. Lovász, Applications of product coloring, Acta Math. Acad. Sci. Hungar. 25(1974), 335–340.CrossRefzbMATHMathSciNetGoogle Scholar
  31. [32]
    A. Gyárfás, T. Jensen, M. Stiebitz, On graphs with strongly independent colour-classes, JGT 46 (2004), 1–14.CrossRefzbMATHGoogle Scholar
  32. [33]
    A. Hajnal, J. Pach, Monochromatic paths in infinite graphs, In: Finite and Infinite Sets, Coll. Math. Soc. J. Bolyai, Eger, 1981, 359–369.Google Scholar
  33. [34]
    A. Harutyunyan, P. M. Kayll, B. Mohar, L. Rafferty, Uniquely D-colourable Digraphs with Large Girth, Canad. J. Math. Vol. 64 (6), 2012, 1310–1328.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [35]
    H. Hatami, Random cubic graphs are not homomorphic to the cycle of size 7, J. Comb. Th. B 93 (2005), 319–325.CrossRefzbMATHMathSciNetGoogle Scholar
  35. [36]
    P. Hell, J. Nešetřil, Graphs and homomorphisms, Oxford University Press, 2004.Google Scholar
  36. [37]
    P. Hell, J. Nešetřil, Colouring, Constraint Satisfaction, and Complexity, Comp. Sci. Review 2,3 (2008) 134–164.Google Scholar
  37. [38]
    S. Hoory, N. Linial, A. Widgerson, Expander graphs and their applications, Bulletin (New series) of the American Mathematical Society 43 (4), 2006, 439–561.CrossRefzbMATHGoogle Scholar
  38. [39]
    S. Janson, T. Luczak, A. Ruczinski, Random Graphs, Wiley, 2000.Google Scholar
  39. [40]
    P. Jeavons, On the algebraic structure of combinatorial problems, Theor. Comp. Sci 200 (1998), 185–204.CrossRefzbMATHMathSciNetGoogle Scholar
  40. [41]
    J. Kahn, Recent results on some not-so-recent hypergraph matching and covering problems, Proc. 1st Int’l Conference on Extremal Problems for Finite Sets, Visegrad, 1991.Google Scholar
  41. [42]
    A. Kechris, Pestov, S. Todorcevic, Fraïssé limits, Ramsey theory and topological dynamics of automorphism groups, Geometrical and Functional Analysis 15(1), 2005, 106–189.CrossRefzbMATHGoogle Scholar
  42. [43]
    J. Kelly and L. Kelly, Path and circuits in critical graphs, Amer. J. Math. 76:786–792, 1954.CrossRefzbMATHMathSciNetGoogle Scholar
  43. [44]
    J. H. Kim, The Ramsey Number R(3,t) has order of magnitude t 2/log t, Random Structures and Algorithms 7 (1995), 173–207.CrossRefzbMATHMathSciNetGoogle Scholar
  44. [46]
    A. V. Kostochka J. Nešetřil, Properties of Descartes’ construction of triangle-free graphs with high chromatic number, Combin. Probab. Comput. 8(1999), no. 5, 467–472.CrossRefzbMATHMathSciNetGoogle Scholar
  45. [47]
    A. Kostochka, J. Nešetřil, P. Smolíková, Coloring and homomorphism of degenerate and bounded degree graphs, Discrete Math. 233, 1–3 (2001), 257–276.CrossRefzbMATHMathSciNetGoogle Scholar
  46. [48]
    A. V. Kostochka, V. Rödl, Constructions of Sparse Uniform Hypergraphs with High Chromatic Number, Random tructures and Algorithms, 2009, 46–56.Google Scholar
  47. [49]
    I. Kříž, A hypergraph free construction of highly chromatic graphs without short cycles, Combinatorica 9 (1989), 227–229.CrossRefzbMATHMathSciNetGoogle Scholar
  48. [50]
    I. Kříž, J. Nešetřil, Chromatic number of the Hasse diagrams, eyebrows and dimension, Order 8 /1991/, 41–48.CrossRefzbMATHMathSciNetGoogle Scholar
  49. [51]
    G. Kun, Constraints, MMSNP and expander relational structures, arXiv: 0706.1701 (2007).Google Scholar
  50. [52]
    G. Kun, M. Szegedy, A new line of attack on the dichotomy conjecture, STOC 2009, 725–734. See also Electronic Coll. on Comp. Compl. (ECCC) 16:59 (2009), 44p.Google Scholar
  51. [53]
    B. Larose, C. Tardif, Graph coloring problems, Wiley, 1995.Google Scholar
  52. [54]
    J. Lenz, D. Mubayi, The poset of hypergraph quasirandomness, The Poset of Hypergraph Quasirandomness, arXiv:1208.5978 au][math.CO] 29 Aug 2012.Google Scholar
  53. [55]
    L. Lovász, On chromatic number of finite set-systems, Acta Math. Acad. Sci. Hungar. 19 (1968), 59–67.CrossRefzbMATHMathSciNetGoogle Scholar
  54. [56]
    L. Lovász, Kneser’s conjecture, chromatic number, and homotopy, J. Combin. Theory. Ser. A 25 (1978), 319–324.CrossRefzbMATHMathSciNetGoogle Scholar
  55. [57]
    L. Lovász, Combinatorial Problems and Exercises, Akad. Kiad, Budapest, 1979.Google Scholar
  56. [58]
    A. Lubotzky, R. Phillips, P. Sarnak, Ramanujan graphs, Combinatorica, 8(3), 1988, 261–277.CrossRefzbMATHMathSciNetGoogle Scholar
  57. [59]
    M. Mares, The saga of minimum spanning trees, Comp. Sci. Review 2 (2008), 165–221.CrossRefGoogle Scholar
  58. [60]
    G. A. Margulis, Explicit constructions of graphs without short cycles and low density codes, Problemy Pereači Informacii, 9(4), 1973, 71–80.zbMATHMathSciNetGoogle Scholar
  59. [61]
    J. Matoušek, Using the Borsuk-Ulam theorem, Lectures on topological methods in combinatorics and geometry, Springer, 2003.Google Scholar
  60. [63]
    M. Molloy, B. Reed, Graph colouring and the probabilistic method, Algorithms and Combinatorics, 23, Springer, 2002.Google Scholar
  61. [64]
    V. Müller, On colorable critical and uniquely colorable critical graphs, in: Recent Advances in Graph Theory (ed. M. Hiedler), Academia, Prague, 1975.Google Scholar
  62. [65]
    V. Müller, On coloring of graphs without short cycles, Discrete Math., 26(1979), 165–179.CrossRefzbMATHMathSciNetGoogle Scholar
  63. [66]
    J. Mycielski, Sur le coloriage des graphes, Colloq. Math. 3, 161–162, 1955.zbMATHMathSciNetGoogle Scholar
  64. [67]
    J. Nešetřil, K-chromatic graphs without cycles of length ≤ 7, (in Russian), Comment. Math. Univ. Carol., 7, 3 (1966), 373–376.zbMATHGoogle Scholar
  65. [68]
    J. Nešetřil, On uniquely colorable graphs without short cycles, Časopis Pěst. Mat. 98 (1973), 122–125.zbMATHGoogle Scholar
  66. [69]
    J. Nešetřil, For graphs there are only four types of hereditary Ramsey classes, J. Combin. Theory Ser. B 46, (1989), no. 2, 127–132.CrossRefzbMATHMathSciNetGoogle Scholar
  67. [70]
    J. Nešetřil, Ramsey Theory, In: Handbook of Combinatorics (ed. R. L. Graham, M. Grötschel, L. Lovász), North-Holland, 1995, 1331–1403.Google Scholar
  68. [71]
    J. Nešetřil, P. Osssona de Mendez, Sparsity — Graph, Structures, and Algorithms, Springer, 2012.Google Scholar
  69. [72]
    J. Nešetřil, V. Rödl, On a probabilistic graph-theoretic Method, Proc. Amer. Math. Soc. 72 (1978), 417–421.zbMATHMathSciNetGoogle Scholar
  70. [73]
    J. Nešetřil, V. Rödl, A short proof of the existence of restricted Ramsey graphs by means of a partite construction, Combinatorica 1, 2 (1982), 199–202.Google Scholar
  71. [74]
    J. Nešetřil, V. Rödl, Sparse Ramsey graphs, Combinatorica 4, 1 (1984), 71–78.CrossRefzbMATHMathSciNetGoogle Scholar
  72. [75]
    J. Nešetřil, V. Rödl, Combinatorial partitions of finite posets and lattices — Ramsey lattices, Algebra Univ. 19 (1984), 106–119CrossRefGoogle Scholar
  73. [76]
    J. Nešetřil, V. Rödl, Complexity of diagrams, Order 3 (1987), 321–330.CrossRefzbMATHMathSciNetGoogle Scholar
  74. [77]
    J. Nešetřil, V. Rödl, Chromatically optimalrigid graphs, J. Combin. Th.(B), 46(1989), 133–141.CrossRefzbMATHGoogle Scholar
  75. [78]
    J. Nešetřil, V. Rödl, More on complexity of diagrams, Comm. Math. Univ. Carol. 36,2 (1995), 269–278.zbMATHGoogle Scholar
  76. [80]
    J. Nešetřil, V. Rödl, Partitions of Finite Relational and Set Systems, J. Comb. Th. A 22,3 (1977), 289–312.zbMATHGoogle Scholar
  77. [81]
    J. Nešetřil, V. Rödl, Partition (Ramsey) Theory — a survey, In: Coll. Math. Soc. János Bolyai, 18. Combinatorics, Keszthely 1976, North Holland, 1978, 759–792.Google Scholar
  78. [82]
    J. Nešetřil, M. Siggers, L. Zadori, A Combinatorial Constraint Satisfaction Problem Dichotomy Classication Conjecture, European J. Comb. 31 (1), 2010, 280–296.CrossRefzbMATHGoogle Scholar
  79. [83]
    J. Nešetřil, X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B 90(2004), no. 1, 161–172.CrossRefzbMATHMathSciNetGoogle Scholar
  80. [84]
    V. Rödl, On the chromatic number of subgraphs of a given graph, Proc. Amer. Math. Soc. 64 (1977), 370–371.CrossRefzbMATHMathSciNetGoogle Scholar
  81. [85]
    V. Rödl, On a packing and covering problem, Europ. J. Combinatorics 5 (1985), 69–78.CrossRefGoogle Scholar
  82. [86]
    V. Rödl, L. Thoma, The complexity of cover graph recognition for some vertices of finite lattices, Order 12,4 (1995), 351–374.CrossRefzbMATHMathSciNetGoogle Scholar
  83. [87]
    G. Simonyi and G. Tardos, Local chromatic number, Ky Fan’s theorem, and circular colorings, Combinatorica 26 (2006), 587–620.CrossRefzbMATHMathSciNetGoogle Scholar
  84. [88]
    J. Spencer, Ten Lectures on the Probabilistic Method, Society for Industrial and Applied Mathematics, 1987.Google Scholar
  85. [90]
    I. M. Wanless, N. C. Wormald, Regular graphs with no homomorphisms onto cycles, J. Comb. Th. B 82 (2001), 155–160.CrossRefzbMATHMathSciNetGoogle Scholar
  86. [91]
    X. Zhu, Uniquely H-colorable graphs with large girth, J. Graph Theory, 23 (1996), 33–41.CrossRefzbMATHMathSciNetGoogle Scholar
  87. [92]
    A. A. Zykov, On some properties of linear complexes, Mat. Sbornik 24:313–319, 1949. (In Russian).MathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2013

Authors and Affiliations

  • Jaroslav Nešetřil
    • 1
    • 2
  1. 1.Computer Science Institute of Charles UniversityPraha 1Czech Republic
  2. 2.Institute for Theoretical Computer Science (ITI)Charles UniversityPraha 1Czech Republic

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