Skip to main content
SpringerLink
Log in

Universality, Tolerance, Chaos and Order

  • Chapter
An Irregular Mind

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 21))

Abstract

What is the minimum possible number of edges in a graph that contains a copy of every graph on n vertices with maximum degree a most k ? This question, as well as several related variants, received a considerable amount of attention during the last decade. In this short survey we describe the known results focusing on the main ideas in the proofs, discuss the remaining open problems, and mention a recent application in the investigation of the complexity of subgraph containment problems.

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

Access this chapter

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 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai, N. Alon, J. Bruck, R. Cypher, C. T. Ho, M. Naor and E. Szemeredi, Fault tolerant graphs, perfect hash functions and disjoint paths, Proc. 33rd IEEE FOCS, Pittsburgh, IEEE (1992), 693–702.

    Google Scholar 

  2. N. Alon, The strong chromatic number of a graph, Random Structures and Algorithms, 3 (1992), 1–7.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Alon and V. Asodi, Sparse universal graphs, Journal of Computational and Applied Mathematics, 142 (2002), 1–11.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. Alon and M. Capalbo, Sparse universal graphs for bounded degree graphs, Random Structures and Algorithms, 31 (2007), 123–133.

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Alon and M. Capalbo, Optimal universal graphs with deterministic embedding, Proc. of the Nineteenth Annual ACM-SIAM SODA (2008), 373–378.

    Google Scholar 

  6. N. Alon, M. Capalbo, Y. Kohayakawa, V. Rödl, A. Ruciński and E. Szemeredi, Universality and tolerance, Proceedings of the 41st IEEE FOCS (2000), 14–21.

    Google Scholar 

  7. N. Alon, M. Capalbo, Y. Kohayakawa, V. Rödl, A. Ruciński and E. Szemeredi, Near-optimum universal graphs for graphs with bounded degrees, RANDOM-APPROX 2001, 170–180.

    Google Scholar 

  8. N. Alon and Z. Füredi, Spanning subgraphs of random graphs, Graphs and Combinatorics, 8 (1992), 91–94.

    Article  MATH  MathSciNet  Google Scholar 

  9. N. Alon and D. Marx, Sparse balanced partitions and the complexity of subgraph problems, in preparation.

    Google Scholar 

  10. N. Alon, R. Yuster and U. Zwick, Color-coding, J. ACM, 42 (1995), 844–856.

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Babai, F. R. K. Chung, P. Erdős, R. L. Graham and J. Spencer, On graphs which contain all sparse graphs, Ann. Discrete Math., 12 (1982), 21–26.

    MATH  Google Scholar 

  12. S. N. Bhatt, F. Chung, F. T. Leighton and A. Rosenberg, Universal graphs for bounded-degree trees and planar graphs, SIAM J. Disc. Math., 2 (1989), 145–155.

    Article  MATH  MathSciNet  Google Scholar 

  13. S. N. Bhatt and C. E. Leiserson, How to assemble tree machines, in: Advances in Computing Research, F. Preparata, ed., 1984.

    Google Scholar 

  14. S. Butler, Induced-universal graphs for graphs with bounded maximum degree, Graphs and Combinatorics, 25 (2009), 461–468.

    Article  MATH  MathSciNet  Google Scholar 

  15. M. Capalbo, A small universal graph for bounded-degree planar graphs, Proc. SODA, 1999, 150–154.

    Google Scholar 

  16. M. Capalbo and S. R. Kosaraju, Small universal graphs, Proc. STOC, 1999, 741–749.

    Google Scholar 

  17. J. Chen, B. Chor, M. Fellows, X. Huang, D. Juedes, I. Kanj and G. Xia, Tight Lower Bounds for Certain Parameterized NP-hard Problems, Proc. 19th Annual IEEE Conference on Computational Complexity, 2004, 150–160.

    Google Scholar 

  18. J. Chen, X. Huang, I. Kanj, and G. Xia, Linear FPT reductions and computational lower bounds, Proc. 36th Annual ACM STOC, ACM, New York (2004), 212–221.

    Google Scholar 

  19. F. R. K. Chung and R. L. Graham, On universal graphs for spanning trees, Proc. London Math. Soc, 27 (1983), 203–211.

    Article  MATH  MathSciNet  Google Scholar 

  20. F. R. K. Chung, R. L. Graham and N. Pippenger, On graphs which contain all small trees II, Proc. 1976 Hungarian Colloquium on Combinatorics (1978), 213–223.

    Google Scholar 

  21. F. R. K. Chung, A. L. Rosenberg and L. Snyder, Perfect storage representations for families of data structures, SIAM J. Alg. Disc. Methods, 4 (1983), 548–565.

    Article  MATH  MathSciNet  Google Scholar 

  22. V. Chvátal, V. Rödl, E. Szemerédi and W. T. Trotter, The Ramsey number of a graph with a bounded maximum degree, J. Combin. Theory Ser. B. (1983), 239–243.

    Google Scholar 

  23. R.G. Downey and M.R. Fellows, Parameterized Complexity, Springer-Verlag, 1999.

    Google Scholar 

  24. L. Esperet, A. Labourel and P. Ochem, On induced-universal graphs for the class of bounded-degree graphs, Inf. Process. Lett, 108(5) (2008), 255–260.

    Article  MathSciNet  Google Scholar 

  25. J. Friedman and N. Pippenger, Expanding graphs contain all small trees, Combinatorica, 7 (1987), 71–76.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Grohe, The complexity of homomorphism and constraint satisfaction problems seen from the other side, J. ACM, 54 (2007).

    Google Scholar 

  27. P. E. Haxell, On the strong chromatic number, Combinatorics, Probability and Computing, 13 (2004), 857–865.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Hajnal and E. Szemerédi, Proof of a conjecture of Erdős, in: Combinatorial Theory and its Applications, Vol. II (P. Erdős, A. Rényi and V. T. Sós, eds.), Colloq. Math Soc. J. Bolyai 4, North Holland, Amsterdam, 1970, 601–623.

    Google Scholar 

  29. R. Impagliazzo, R. Paturi and F. Zane, Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63 (2001), 512–530.

    Article  MATH  MathSciNet  Google Scholar 

  30. Y. Kohayakawa, V. Rödl, M. Schacht and E. Szemerédi, Sparse partition universal graphs for graphs of bounded degree, to appear.

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. G. A. Margulis, Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and super concentrators, Problems of Information Transmission, 24 (1988), 39–46.

    Google Scholar 

  33. D. Marx, Can you beat treewidth? Proc. 48th Annual IEEE FOCS, (2007), 169–179. Also: Theory of Computing, to appear.

    Google Scholar 

  34. N. Robertson and P. Seymour. Graph minors. II. Algorithmic aspects of tree-width, Journal of Algorithms, 7 (1986), 309–322.

    Article  MATH  MathSciNet  Google Scholar 

  35. V. Rodl, A note on universal graphs, Ars. Combin., 11 (1981), 225–229.

    MATH  MathSciNet  Google Scholar 

  36. V. Rödl, A. Ruciński and A. Taraz, Hypergraph packing and graph embedding, Combinatorics, Probability and Computing, 8 (1999) 363–376.

    Article  MATH  MathSciNet  Google Scholar 

  37. A. Rucinski, Matching and covering the vertices of a random graph by copies of a given graph, Discrete Math., 105 (1992), 185–197.

    Article  MATH  MathSciNet  Google Scholar 

  38. E. Szemerédi, Regular partitions of graphs, in: Proc. Colloque Inter. CNRS (J. C. Bermond, J. C. Fournier, M. Las Vergnas and D. Sotteau, eds.), 1978, 399–401.

    Google Scholar 

  39. D. P. West, Introduction to Graph Theory, Prentice-Hall, 1996.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Schools of Mathematics and Computer Science Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Noga Alon

Authors
  1. Noga Alon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Noga Alon .

Editor information

Editors and Affiliations

  1. Alfred Rènyi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Imre Bárány

  2. Department of Mathematics, University of British Columbia, Mathematics Road 1984, V6T 1Z2, Vancouver, British Columbia, Canada

    József Solymosi

  3. Alfred Rènyi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Gábor Sági

Rights and permissions

Reprints and permissions

Copyright information

© 2010 János Bolyai Mathematical Society and Springer-Verlag

About this chapter

Cite this chapter

Alon, N. (2010). Universality, Tolerance, Chaos and Order. In: Bárány, I., Solymosi, J., Sági, G. (eds) An Irregular Mind. Bolyai Society Mathematical Studies, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14444-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation