Skip to main content
SpringerLink
Log in
Book cover

Ramsey Theory pp 77–96Cite as

Birkhäuser

Large Monochromatic Components in Edge Colorings of Graphs: A Survey

  • Chapter
  • First Online:

Part of the book series: Progress in Mathematics ((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.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. B. Andrásfai, Remarks on a paper of Gerencsér and Gyárfás,Ann. Univ. Sci. Eötvös, Budapest 13(1970) 103–107.

    Google Scholar 

  2. A. Bialostocki, P. Dierker, Zero sum Ramsey theorems,Congr. Numer. 70(1990) 119–130.

    Google Scholar 

  3. A. Bialostocki, P. Dierker, W. Voxman, Either a graph or its complement is connected: a continuing saga, unpublished manuscript.

    Google Scholar 

  4. A. Bialostocki, W. Voxman, On monochromatic-rainbow generalizations of two Ramsey type theorems,Ars Combinatoria 68(2003) 131–142.

    Google Scholar 

  5. J. Bierbrauer, A. Brandis, On generalized Ramsey numbers for trees,Combinatorica 5(1985) 95–107.

    Google Scholar 

  6. J. Bierbrauer, A. Gyárfás, On (n,k)-colorings of complete graphs,Congr. Numer. 58(1987) 127–139.

    Google Scholar 

  7. J. Bierbrauer, Weighted arcs, the finite Radon transform and a Ramsey problem,Graphs Combin. 7(1991) 113–118.

    Google Scholar 

  8. B. Bollobás,Modern Graph Theory,Springer, Berlin (1998).

    Book  MATH  Google Scholar 

  9. B. Bollobás, A. Gyárfás, Highly connected monochromatic subgraphs,Discrete Math. 308(2008) 1722–1725.

    Google Scholar 

  10. A. A. Bruen, M. de Resmini, Blocking sets in affine planes,Ann. Discrete Math. 18(1983) 169–176.

    Google Scholar 

  11. S. A. Burr, Either a graph or its complement contains a spanning broom, unpublished manuscript, 1992.

    Google Scholar 

  12. K. Cameron, J. Edmonds, Lambda composition,J. Graph Theor. 26(1997) 9–16.

    Google Scholar 

  13. E. J. Cockayne, P. Lorimer, The Ramsey numbers for stripes,J. Austral. Math. Soc. Ser, A. 19(1975) 252–256.

    Google Scholar 

  14. P. Erdős, T. Fowler, Finding large p-colored diameter two subgraphs,Graphs Combin. 15(1999) 21–27.

    Google Scholar 

  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. 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. Z. Füredi, A. Gyárfás, Coveringt-element sets by partitions,Europ. J. Combin. 12(1991) 483–489.

    Google Scholar 

  18. Z. Füredi, D. J. Kleitman, On zero-trees,J. Graph Theor. 16(1992) 107–120.

    Google Scholar 

  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. Z. Füredi, Maximum degree and fractional matchings in uniform hypergraphs,Combinatorica 1(1981) 155–162.

    Google Scholar 

  21. Z. Füredi, Covering the complete graph by partitions,Discrete Math. 75(1989) 217–226.

    Google Scholar 

  22. Z. Füredi, Intersecting designs from linear programming and graphs of diameter two,Discrete Math. 127(1993) 187–207.

    Google Scholar 

  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. L. Gerencsér, A. Gyárfás, On Ramsey type problems,Ann. Univ. Sci. Eötvös, Budapest 10(1967) 167–170.

    Google Scholar 

  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. A. Gyárfás, Fruit salad,Electron. J. Combon. 4(1997) R8.

    Google Scholar 

  27. A. Gyárfás, Ramsey and Turán-type problems in geometric bipartite graphs,Electron. Notes Discrete Math, 31(2008) 253–254.

    Article  Google Scholar 

  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. A. Gyárfás, P. Haxell, Large monochromatic components in colorings of complete hypergraphs,Discrete Math. 309(2009) 3156–3160.

    Google Scholar 

  30. A. Gyárfás, M. Ruszinkó, G. N. Sárközy, E. Szemerédi, Three-color Ramsey numbers for paths,Combinatorica 27(2007) 35–69.

    Google Scholar 

  31. A. Gyárfás, G. Simonyi, Edge colorings of complete graphs without tricolored triangles,J.Graph Theor. 46(2004) 211–216.

    Google Scholar 

  32. A. Gyárfás, G. N. Sárközy, Size of monochromatic components in local edge colorings,Discrete Math. 308(2008) 2620–2622.

    Google Scholar 

  33. A. Gyárfás, G. N. Sárközy, Size of monochromatic double stars in edge colorings,Graphs Combin. 24(2008) 531–536.

    Google Scholar 

  34. A. Gyárfás, G. N. Sárközy, Gallai-colorings on non-complete graphs,Discrete Math. 310(2010) 977–980.

    Article  MathSciNet  MATH  Google Scholar 

  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. A. Gyárfás, G. N. Sárközy, E. Szemerédi, Stability of the path-path Ramsey number,Discrete Math. 309(2009) 4590–4595.

    Google Scholar 

  37. R. E. Jamison, Covering finite fields with cosets of subspaces,J. Combin. Theor A. 22(1977) 253–266.

    Google Scholar 

  38. M. Kano, X. Li, Monochromatic and heterochromatic subgraphs in edge-colored graphs – a survey,Graphs Combin. 24(2008) 237–263.

    Google Scholar 

  39. G. Károlyi, J. Pach, G. Tóth, Ramsey-type results for geometric graphs, I.,Discrete Comput. Geom. 18(1997) 247–255.

    Google Scholar 

  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. H. Liu, R. Morris, N. Prince, Highly connected monochromatic subgraphs of multicoloured graphs, I, to appear inJ. Graph Theor.

    Google Scholar 

  42. H. Liu, R. Morris, N. Prince, Highly connected monochromatic subgraphs of multicoloured graphs, II.Discrete Math. 308(2008) 5096–5121.

    Google Scholar 

  43. L. Lovász,Combinatorial Problems and Exercises,North-Holland, Amsterdam (1993).

    MATH  Google Scholar 

  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. D. Mubayi, Generalizing the Ramsey problem through diameter,Electron. J. Combin. 9(2002) R41.

    Google Scholar 

  46. J. Pach, P. K. Agarwal,Combin. Geomet.Wiley, New York (1995).

    Google Scholar 

  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. Zs. Tuza, On special cases of Ryser’s conjecture, unpublished manuscript.

    Google Scholar 

  49. D. West,Introduction to Graph Theory, Prentice Hall, Englewood Cliffs, NJ, 2000.

    Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Computer and Automation Research Institute, Hungarian Academy of Sciences, Budapest, 63, Budapest, 1518, Hungary

    András Gyárfás

Authors
  1. András Gyárfás
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to András Gyárfás .

Editor information

Editors and Affiliations

  1. at Colorado Springs, College of Letters, Arts, and Sciences, University of Colorado, Austin Bluffs Parkway 1420, Colorado Springs, 80918, Colorado, USA

    Alexander Soifer

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Gyárfás, A. (2011). Large Monochromatic Components in Edge Colorings of Graphs: A Survey. In: Soifer, A. (eds) Ramsey Theory. Progress in Mathematics, vol 285. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8092-3_5

Download citation

Publish with us

Policies and ethics

Search

Navigation