Advertisement

Perfect colorings of the Johnson graphs J(8, 3) and J(8, 4) with two colors

  • S. V. Avgustinovich
  • I. Yu. Mogil’nykh
Article

Abstract

In this article the parameter matrices are enumerated of all perfect 2-colorings of the Johnson graphs J(8, 3) and J(8, 4), and several constructions are presented for perfect 2-coloring of J(2w, w) and J(2m, 3). The concept of a perfect coloring generalizes the concept of completely regular code introduced by P. Delsarte. The problem of existence of similar structures in Johnson graphs is closely related to the problem of existence of completely regular codes in Johnson graphs and, in particular, to the Delsarte conjecture on the nonexistence of nontrivial perfect codes in Johnson graphs, the problem of existence of block designs, and other well-known problems.

Keywords

perfect coloring Johnson scheme block design 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    I. Yu. Mogilnykh, “On the Regularity of Perfect 2-Colorings of the Johnson Graph,” Problemy Peredachi Inform. 43(4), 37–44 (2007) [Problems Inform. Transmission 43 (4), 303–309 (2007)].MathSciNetGoogle Scholar
  2. 2.
    I. Yu. Mogilnykh, “On Nonexistence of Some Perfect 2-Colorings of Johnson Graphs,” Diskret. Anal. Issled. Oper. 16(5), 52–68 (2009).MathSciNetGoogle Scholar
  3. 3.
    D. Tsvetkovich, M. Dub, and Kh. Zakhs, Spectra of Graphs: Theory and Application (Naukova Dumka, Kiev, 1984) [in Russian].Google Scholar
  4. 4.
    D. G. Fon-der-Flaass, “Perfect 2-Colorings of a 12-Dimensional Cube That Achieve a Bound of Correlation Immunity,” Siberian Electron.Mat. Izv. No. 4, 292–295 (2007).Google Scholar
  5. 5.
    D. G. Fon-der-Flaass, “Perfect 2-Colorings of a Hypercube,” Sibirsk. Mat. Zh. 48(4), 923–930 (2007) [SiberianMath. J. 48 (4), 740–745 (2007)].MathSciNetMATHGoogle Scholar
  6. 6.
    M. Dehon, “On the Existence of 2-Designs S λ(2, 3, υ) Without Repeated Blocks,” DiscreteMath. 43, 155–171 (1961).MathSciNetCrossRefGoogle Scholar
  7. 7.
    P. Delsarte, “An AlgebraicApproach to the Association Schemes of Coding Theory,” Philips Res. Rep. Suppl. 10, 1–97 (1973).MathSciNetGoogle Scholar
  8. 8.
    T. Etzion and M. Schwarz, “Perfect Constant-Weight Codes,” IEEE Trans. Inform. Theory. 50(9), 2156–2165 (2004).MathSciNetCrossRefGoogle Scholar
  9. 9.
    D. G. Fon-der-Flaass, “A Bound on Correlation Immunity,” Siberian Electron. Mat. Izv. No. 4, 133–135 (2007).Google Scholar
  10. 10.
    C. Godsil and G. Royle, Algebraic Graph Theory (Springer, New York, 2001).MATHGoogle Scholar
  11. 11.
    H. Hanani, “On Quadruple Systems,” Can. J. Math. 12, 145–157 (1960).MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    A. Hartman and K. T. Phelps, “Tetrahedral Quadruple Systems,” Utilitas Math. 37, 181–189 (1990).MathSciNetMATHGoogle Scholar
  13. 13.
    W. J. Martin, “Completely Regular Designs of Strength One,” J. Algebr. Comb. 3, 17–185 (1994).CrossRefGoogle Scholar
  14. 14.
    W. J. Martin, “Completely Regular Designs,” J. Comb. Des. 6(4), 261–273 (1998).MATHCrossRefGoogle Scholar
  15. 15.
    A. Meyerowitz, “Cycle-Balanced Partitions in Distance-Regular Graphs,” Discrete Math. 264, 149–165 (2003).MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    I. Yu. Mogilnykh and S. V. Avgustinovich, “Perfect Colorings with Two Colors of a Johnson Graph,” in Proceedings of the XI International Symposium on Problems of Redundancy in Information and Control Systems (St. Petersburg, Russia, July 2–6, 2007) (Saint Petersburg State Univ. of Aerospace Instrumentation, St. Petersburg, 2007), pp. 205–209.Google Scholar
  17. 17.
    I. Yu. Mogilnykh and S. V. Avgustinovich, “Perfect 2-Colorings of Johnson Graphs J(6, 3) and J(7, 3),” in Lecture Notes in Computer Science, Vol. 5228 (Springer, 2008), pp. 11–19.CrossRefGoogle Scholar
  18. 18.
    K. T. Phelps, D. R. Stinson, and S. A. Vanstone, “The Existence of Simple S 3(3, 4, υ),” DiscreteMath. 77, 255–258 (1989).MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    G. S. Shapiro and D. S. Slotnik, “On the Mathematical Theory of Error Correcting Codes,” IBM J. Res. Dev. 3, 68–72 (1959).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia

Personalised recommendations