Skip to main content
SpringerLink
Log in

A Generalization of Kneser Graphs

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

Graphs which are analogs of Kneser graphs are studied. The problem of determining the chromatic numbers of these graphs is considered. It is shown that their structure is similar to that of Kneser graphs. Upper and lower bounds for the chromatic numbers of the graphs under examination are obtained. For certain parameter values, an order-sharp estimate of the chromatic numbers is found, and in some cases, the exact value of the quantity in question is determined.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Kneser, “Aufgabe 300,” Jahresber. Deutsch. Math. Verein 58, 27 (1955).

    Google Scholar 

  2. L. Lovasz, “Kneser’s conjecture, chromatic number, and homotopy,” J. Combin. Theory Ser. A 25 (3), 319–324 (1978).

    Article  MathSciNet  Google Scholar 

  3. J. Matousek, Using the Borsuk—Ulam Theorem (Springer, Berlin, 2003).

    MATH  Google Scholar 

  4. A. M. Raigorodskii, “Combinatorial geometry and coding theory,” Fund. Inform. 145 (3), 359–369 (2016).

    Article  MathSciNet  Google Scholar 

  5. A. M. Raigorodskii, Kneser’s Conjecture and the Topological Method in Combinatorics (MTsNMO, Moscow, 2011) [in Russian].

    Google Scholar 

  6. S. G. Kiselev and A. M. Raigorodskii, “On the chromatic number of a random subgraph of the Kneser graph,” Dokl. Akad. Nauk 476 (4), 375–376 (2017) [Dokl. Math. 96 (2), 475–476 (2017)].

    MathSciNet  MATH  Google Scholar 

  7. M. M. Pyaderkin and A. M. Raigorodskii, “On random subgraphs of Kneser graph and their generalizations,” Dokl. Akad. Nauk 470 (4), 384–386 (2016) [Dokl. Math. 94 (2), 547–549 (2016)].

    MATH  Google Scholar 

  8. A. M. Raigorodskii, “Lovasz theorem on the chromatic number of spheres revisited,” Mat. Zametki 98 (3), 470–471 (2015) [Math. Notes 98 (3), 522–524 (2015)].

    Article  MathSciNet  Google Scholar 

  9. A. Kupavskii, “On random subgraphs of Kneser and Schrijver graphs,” J. Combin. Theory Ser. A 141 (3), 8–15 (2016).

    Article  MathSciNet  Google Scholar 

  10. A. Kupavskii, “Random Kneser graphs and hypergraphs,” Electron. J. Combin. 25 (4), 4–52 (2018).

    Article  MathSciNet  Google Scholar 

  11. A. Kupavskii, Sharp Bounds for the Chromatic Number of Random Kneser Graphs, ArXiv: 1810.01161 (2018).

    Google Scholar 

  12. J. Balogh, D. Cherkashin, and S. Kiselev, Coloring General Kneser Graphs and Hypergraphs via High-Discrepancy Hypergraphs, ArXiv: 1805.09394 (2018).

    MATH  Google Scholar 

  13. A. V. Bobu and A. E. Kupriyanov, “On chromatic numbers of close-to-Kneser distance graphs,” Problemy Peredachi Informatsii 52 (4), 64–83 (2016) [Problems Inform. Transmission 52 (4), 373–390 (2016)].

    MathSciNet  MATH  Google Scholar 

  14. P. Frankl and R. M. Wilson, “Intersection theorems with geometric consequences,” Combinatorica 1 (4), 357–368 (1981).

    Article  MathSciNet  Google Scholar 

  15. A. M. Raigorodskii, “Borsuk’s problem and the chromatic numbers of some metric spaces,” Uspekhi Mat. Nauk 56 (1), 107–146 (2001) [Russian Math. Surveys 56 (1), 103–139 (2001)].

    Article  MathSciNet  Google Scholar 

  16. A. M. Raigorodskii, “On the chromatic numbers of spheres inRn,” Combinatorica 32 (1), 111–123 (2012).

    Article  MathSciNet  Google Scholar 

  17. A. B. Kupavskii, “On the colorings of spheres embedded inRn,” Mat. Sb. 202 (6), 83–110 (2011) [Sb. Math. 202 (6), 859–886 (2011)].

    Article  MathSciNet  Google Scholar 

  18. O. A. Kostina and A. M. Raigorodskii, “On lower bounds for the chromatic number of sphere,” Dokl. Akad. Nauk 463 (6), 639 (2015) [Dokl. Math. 92 (1), 500–502 (2015)].

    MathSciNet  MATH  Google Scholar 

  19. D. Cherkashin, A. Kulikov, and A. Raigorodskii, “On the chromatic numbers of small-dimensional Euclidean spaces,” Discrete Appl. Math. 243, 125–131 (2018).

    Article  MathSciNet  Google Scholar 

  20. A. M. Raigorodskii and D. D. Cherkashin, “On the chromatic numbers of low-dimensional spaces,” Dokl. Akad. Nauk 472 (1), 11–12 (2017) [Dokl. Math. 95 (1), 5–6(2017)].

    MATH  Google Scholar 

  21. A. M. Raigorodskii and A. A. Sagdeev, “On a bound in extremal combinatorics,” Dokl. Akad. Nauk 478 (3), 271–273 (2018) [Dokl. Math. 97 (1), 47–48 (2018)].

    MathSciNet  MATH  Google Scholar 

  22. E. I. Ponomarenko and A. M. Raigorodskii, “New upper bounds for the independence numbers of graphs with vertices in {- 1,0,1}n and their applications to problems of the chromatic numbers of distance graphs,” Mat. Zametki 96 (1), 138–147 (2014) [Math. Notes 96 (1), 140–148 (2014)].

    Article  MathSciNet  Google Scholar 

  23. A. V. Bobu and A. E. Kupriyanov, “Refinement of lower bounds of the chromatic number of a space with forbidden one-color triangles,” Mat. Zametki 105 (3), 349–363 (2019) [Math. Notes 105 (3), 329–341 (2019)].

    Article  MathSciNet  Google Scholar 

  24. A. V. Bobu, A. E. Kupriyanov, and A. M. Raigorodskii, “On the number of edges of a uniform hypergraph with a range of allowed intersections,” Problemy Peredachi Informatsii 53 (4), 16–42 (2017) [Problems Inform. Transmission 53 (4), 319–342 (2017)].

    MathSciNet  MATH  Google Scholar 

  25. A. M. Raigorodskii, A. V. Bobu, and A. E. Kupriyanov, “On the number of edges of a uniform hypergraph with a range of permitted intersections,” Dokl. Akad. Nauk 475 (4), 365–368 (2017) [Dokl. Math. 96 (1), 354–357 (2017)].

    MathSciNet  MATH  Google Scholar 

  26. A. V. Bobu, A. E. Kupriyanov, and A. M. Raigorodskii, “Asymptotic study of the maximum number of edges in a uniform hypergraph with one forbidden intersection,” Mat. Sb. 207 (5), 17–42 (2016) [Sb. Math. 207 (5), 652–677 (2016)].

    Article  MathSciNet  Google Scholar 

  27. P. Frankl and A. Kupavskii, “Erdos-Ko-Rado theorem for {0, ±1}-vectors,” J. Combin. Theory Ser. A 155, 157–179 (2018).

    Article  MathSciNet  Google Scholar 

  28. P. Frankl and A. Kupavskii, “Intersection theorems for {0, ±1}-vectors and s-cross-intersecting families,” Mosc. J. Comb. Number Theory 7 (2), 2–21 (2017).

    MathSciNet  MATH  Google Scholar 

  29. P. Frankl and A. Kupavskii, “Families of vectors without antipodal pairs,” Studia Sci. Math. Hungar. 55 (2), 231–237 (2018).

    MathSciNet  MATH  Google Scholar 

  30. A. M. Raigorodskii and E. A. Shishunov, “On the independence numbers of distance graphs with vertices in {- 1,0,1}™,” Dokl. Akad. Nauk 488 (5), 486–487 (2019) [Dokl. Math. 100 (2), 476–477 (2019)].

    MATH  Google Scholar 

  31. A. M. Raigorodskii and T. V. Trukhan, “On the chromatic numbers of some distance graphs,” Dokl. Akad. Nauk 482 (6), 648–650 (2018) [Dokl. Math. 98 (2), 515–517 (2018)].

    MATH  Google Scholar 

  32. A. Raigorodskii and D. Cherkashin, “Extremum problems in hypergraph colorings,” Uspekhi Mat. Nauk 75 (1), 95–154 (2020).

    MathSciNet  Google Scholar 

  33. P. Erdos, Ch. Ko, and R. Rado, “Intersection theorems for systems of finite sets,” Quart. J. Math. Oxford Ser. (2) 12, 313–320 (1961).

    Article  MathSciNet  Google Scholar 

  34. A. Kupavskii, “Diversity of uniform intersecting families,” European J. Combin. 74, 39–47 (2018).

    Article  MathSciNet  Google Scholar 

  35. P. Frankl and A. Kupavskii, “New inequalities for families without k pairwise disjoint members,” J. Combin. Theory Ser. A 157, 427–434 (2018).

    Article  MathSciNet  Google Scholar 

  36. A. Kupavskii and D. Zakharov, “Regular bipartite graphs and intersecting families,” J. Combin. Theory Ser. A 155, 180–189 (2018).

    Article  MathSciNet  Google Scholar 

  37. P. Frankl and A. Kupavskii, “Counting intersecting and pairs of cross-intersecting families,” Combin. Probab. Comput. 27 (1), 60–68 (2018).

    Article  MathSciNet  Google Scholar 

  38. P. Frankl and A. Kupavskii, “A size-sensitive inequality for cross-intersecting families,” European J. Combin. 62, 263–271 (2017).

    Article  MathSciNet  Google Scholar 

  39. P. Frankl and A. Kupavskii, “Uniform s-cross-intersecting families,” Combin. Probab. Comput. 26 (4), 517–524 (2017).

    Article  MathSciNet  Google Scholar 

  40. B. Bollobás, B. P. Narayanan, and A. M. Raigorodskii, “On the stability of the Erdos-Ko-Rado theorem,” J. Combin. Theory Ser. A 137, 64–78 (2016).

    Article  MathSciNet  Google Scholar 

Download references

Funding

This work was supported by the Russian Foundation for Basic Research under grant 18-01-00355 and by the Presidential Program for the State Support of Leading Scientific Schools under grant NSh-6760.2018.1.

Author information

Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    A. V. Bobu, A. É. Kupriyanov & A. M. Raigorodskii

  2. Moscow Institute of Physics and Technology (State University), Dolgoprudnyi, Moscow Oblast, 141701, Russia

    A. M. Raigorodskii

  3. Caucasus Mathematical Center, Adyghe State University, Maikop, 352700, Russia

    A. M. Raigorodskii

  4. Institute for Mathematics and Informatics, Buryat State University, Ulan-Ude, 670000, Russia

    A. M. Raigorodskii

Authors
  1. A. V. Bobu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. É. Kupriyanov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. M. Raigorodskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. V. Bobu, A. É. Kupriyanov or A. M. Raigorodskii.

Additional information

Russian Text © The Author(s), 2020, published in Matematicheskie Zametki, 2020, Vol. 107, No. 3, pp. 351–365.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bobu, A.V., Kupriyanov, A.É. & Raigorodskii, A.M. A Generalization of Kneser Graphs. Math Notes 107, 392–403 (2020). https://doi.org/10.1134/S0001434620030037

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0001434620030037

Keywords

Search

Navigation