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Large Monochromatic Components in Edge Colorings of Graphs: A Survey

  • András Gyárfás
Chapter
Part of the Progress in Mathematics book series (PM, volume 285)

Abstract

The aim of this survey is to summarize an area of combinatorics that lies on the border of several areas: Ramsey theory, resolvable block designs, factorizations, fractional matchings and coverings, and partition covers.

Keywords

Complete Graph Complete Bipartite Graph Color Class Edge Coloring Geometric Graph 
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.

Notes

Acknowledgement

Thanks to Arie Bialostocki for his inspiring conjectures and for his remarks on this survey. The careful reading of a referee is also appreciated. Research supported in part by OTKA Grant No. K68322.

References

  1. [1]
    B. Andrásfai, Remarks on a paper of Gerencsér and Gyárfás,Ann. Univ. Sci. Eötvös, Budapest 13(1970) 103–107.Google Scholar
  2. [2]
    A. Bialostocki, P. Dierker, Zero sum Ramsey theorems,Congr. Numer. 70(1990) 119–130.Google Scholar
  3. [3]
    A. Bialostocki, P. Dierker, W. Voxman, Either a graph or its complement is connected: a continuing saga, unpublished manuscript.Google Scholar
  4. [4]
    A. Bialostocki, W. Voxman, On monochromatic-rainbow generalizations of two Ramsey type theorems,Ars Combinatoria 68(2003) 131–142.Google Scholar
  5. [5]
    J. Bierbrauer, A. Brandis, On generalized Ramsey numbers for trees,Combinatorica 5(1985) 95–107.Google Scholar
  6. [6]
    J. Bierbrauer, A. Gyárfás, On (n,k)-colorings of complete graphs,Congr. Numer. 58(1987) 127–139.Google Scholar
  7. [7]
    J. Bierbrauer, Weighted arcs, the finite Radon transform and a Ramsey problem,Graphs Combin. 7(1991) 113–118.Google Scholar
  8. [8]
    B. Bollobás,Modern Graph Theory,Springer, Berlin (1998).MATHCrossRefGoogle Scholar
  9. [9]
    B. Bollobás, A. Gyárfás, Highly connected monochromatic subgraphs,Discrete Math. 308(2008) 1722–1725.Google Scholar
  10. [10]
    A. A. Bruen, M. de Resmini, Blocking sets in affine planes,Ann. Discrete Math. 18(1983) 169–176.Google Scholar
  11. [11]
    S. A. Burr, Either a graph or its complement contains a spanning broom, unpublished manuscript, 1992.Google Scholar
  12. [12]
    K. Cameron, J. Edmonds, Lambda composition,J. Graph Theor. 26(1997) 9–16.Google Scholar
  13. [13]
    E. J. Cockayne, P. Lorimer, The Ramsey numbers for stripes,J. Austral. Math. Soc. Ser, A. 19(1975) 252–256.Google Scholar
  14. [14]
    P. Erdős, T. Fowler, Finding large p-colored diameter two subgraphs,Graphs Combin. 15(1999) 21–27.Google Scholar
  15. [15]
    P. Erdős, R. Faudree, A. Gyárfás, R. H. Schelp, Domination in colored complete graphs,J.Graph Theor. 13(1989) 713–718.Google Scholar
  16. [16]
    R. J. Faudree, R. J. Gould, M. Jacobson, L. Lesniak, Complete families of graphs,Bull. Inst. Comb. Appl. 31(2001) 39–44.Google Scholar
  17. [17]
    Z. Füredi, A. Gyárfás, Coveringt-element sets by partitions,Europ. J. Combin. 12(1991) 483–489.Google Scholar
  18. [18]
    Z. Füredi, D. J. Kleitman, On zero-trees,J. Graph Theor. 16(1992) 107–120.Google Scholar
  19. [19]
    A. Figaj, T. Łuczak, Ramsey numbers for a triple of long even cycles,J. Combin. Theor. B 97(2007) 584–596.Google Scholar
  20. [20]
    Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs,Combinatorica 1(1981) 155–162.Google Scholar
  21. [21]
    Z. Füredi, Covering the complete graph by partitions,Discrete Math. 75(1989) 217–226.Google Scholar
  22. [22]
    Z. Füredi, Intersecting designs from linear programming and graphs of diameter two,Discrete Math. 127(1993) 187–207.Google Scholar
  23. [23]
    T. Gallai, Transitiv orientierbare Graphen,Acta Math. Sci. Hungar. 18(1967) 25–66. English translation by F. Maffray and M. Preissmann, in: J. L. Ramirez-Alfonsin and B. A. Reed (editors),Perfect Graphs, Wiley, New York (2001) 25–66.Google Scholar
  24. [24]
    L. Gerencsér, A. Gyárfás, On Ramsey type problems,Ann. Univ. Sci. Eötvös, Budapest 10(1967) 167–170.Google Scholar
  25. [25]
    A. Gyárfás, Partition coverings and blocking sets in hypergraphs (in Hungarian)Commun. Comput. Autom. Inst. Hungar. Acad. Sci. 71(1977) 62 pp.Google Scholar
  26. [26]
    A. Gyárfás, Fruit salad,Electron. J. Combon. 4(1997) R8.Google Scholar
  27. [27]
    A. Gyárfás, Ramsey and Turán-type problems in geometric bipartite graphs,Electron. Notes Discrete Math, 31(2008) 253–254.CrossRefGoogle Scholar
  28. [28]
    A. Gyárfás, Ramsey and Turán-type problems in geometric bipartite graphs, to appear in Ann. Sci. Eotvos,Sect. Math. Budapest, 53(2008).Google Scholar
  29. [29]
    A. Gyárfás, P. Haxell, Large monochromatic components in colorings of complete hypergraphs,Discrete Math. 309(2009) 3156–3160.Google Scholar
  30. [30]
    A. Gyárfás, M. Ruszinkó, G. N. Sárközy, E. Szemerédi, Three-color Ramsey numbers for paths,Combinatorica 27(2007) 35–69.Google Scholar
  31. [31]
    A. Gyárfás, G. Simonyi, Edge colorings of complete graphs without tricolored triangles,J.Graph Theor. 46(2004) 211–216.Google Scholar
  32. [32]
    A. Gyárfás, G. N. Sárközy, Size of monochromatic components in local edge colorings,Discrete Math. 308(2008) 2620–2622.Google Scholar
  33. [33]
    A. Gyárfás, G. N. Sárközy, Size of monochromatic double stars in edge colorings,Graphs Combin. 24(2008) 531–536.Google Scholar
  34. [34]
    A. Gyárfás, G. N. Sárközy, Gallai-colorings on non-complete graphs,Discrete Math. 310(2010) 977–980.MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    A. Gyárfás, G. N. Sárközy, A. Sebő, S. Selkow, Ramsey-type results for Gallai-colorings,J.Graph Theor. 64(2009) 233–243.Google Scholar
  36. [36]
    A. Gyárfás, G. N. Sárközy, E. Szemerédi, Stability of the path-path Ramsey number,Discrete Math. 309(2009) 4590–4595.Google Scholar
  37. [37]
    R. E. Jamison, Covering finite fields with cosets of subspaces,J. Combin. Theor A. 22(1977) 253–266.Google Scholar
  38. [38]
    M. Kano, X. Li, Monochromatic and heterochromatic subgraphs in edge-colored graphs – a survey,Graphs Combin. 24(2008) 237–263.Google Scholar
  39. [39]
    G. Károlyi, J. Pach, G. Tóth, Ramsey-type results for geometric graphs, I.,Discrete Comput. Geom. 18(1997) 247–255.Google Scholar
  40. [40]
    G. Károlyi, J. Pach, G. Tóth, P. Valtr, Ramsey-type results for geometric graphs, II.,Discrete Comput. Geom. 20(1998) 375–388.Google Scholar
  41. [41]
    H. Liu, R. Morris, N. Prince, Highly connected monochromatic subgraphs of multicoloured graphs, I, to appear inJ. Graph Theor. Google Scholar
  42. [42]
    H. Liu, R. Morris, N. Prince, Highly connected monochromatic subgraphs of multicoloured graphs, II.Discrete Math. 308(2008) 5096–5121.Google Scholar
  43. [43]
    L. Lovász,Combinatorial Problems and Exercises,North-Holland, Amsterdam (1993).MATHGoogle Scholar
  44. [44]
    T. Łuczak,R(C n,C n,C n)≤(4+o(1))n,J. Combin. Theor. B. 75(1999) 174–187.Google Scholar
  45. [45]
    D. Mubayi, Generalizing the Ramsey problem through diameter,Electron. J. Combin. 9(2002) R41.Google Scholar
  46. [46]
    J. Pach, P. K. Agarwal,Combin. Geomet.Wiley, New York (1995).Google Scholar
  47. [47]
    A. Schrijver, P. D. Seymour, A simpler proof and a generalization of the Zero-trees theorem,J. Combin. Theor. A. 58(1991) 301–305.Google Scholar
  48. [48]
    Zs. Tuza, On special cases of Ryser’s conjecture, unpublished manuscript.Google Scholar
  49. [49]
    D. West,Introduction to Graph Theory, Prentice Hall, Englewood Cliffs, NJ, 2000.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Computer and Automation Research InstituteHungarian Academy of Sciences, BudapestBudapestHungary

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