Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Perfect 3-Colorings of the Johnson Graph J(6, 3)

  • 3 Accesses

Abstract

Under a perfect coloring with m colors (a perfect m-coloring) with matrix \(A=\{a_{ij}\}_{i,j=1,\ldots ,m}\) of a graph G,  we understand a coloring of the vertices G with the colors \(\{1,\ldots ,m\}\) such that the number of vertices of color j adjacent to a fixed vertex of color i is equal to \(a_{ij}\) independently of the choice of the latter vertex. The matrix A is called the parameter matrix of a perfect coloring. This paper introduces the method breaking, the inverse of merging, which produces sufficient constructions for the paper. Furthermore, the parameter matrices of all perfect 3-colorings of the Johnson graph J(6, 3) are enumerated.

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

References

  1. 1.

    Alaeiyan, M.H., Abedi, A.: Perfect \(3\)-colorings of the Johnson graphs \(J(4, 2)\), \(J(5, 2)\), \(J(6, 2)\) and Petersen graph. Ars Combin. 140, 199–213 (2018)

  2. 2.

    Avgustinovich, S.V., Mogilnykh, I.Y.: Perfect colorings of the Johnson graphs \(J(8,3)\) and \(J(8,4)\) with two colors. J. Appl. Ind. Math. 5, 19–30 (2011)

  3. 3.

    Camion, P., Courteau, B., Delsarte, P.: On \(r\)-partition designs in Hamming spaces. Appl. Algebra Eng. Commun. Comput. 2, 147–162 (1992)

  4. 4.

    Gavrilyuk, A.L., Goryainov, S.V.: On perfect \(2\)-colorings of Johnson graphs \(J(\upsilon, 3)\). J. Combin. Des. 21, 232–252 (2013)

  5. 5.

    Krotov, D.S.: On weight distributions of perfect colorings and completely regular codes. Des. Codes Crypt. 61, 315–329 (2011)

  6. 6.

    Martin, W.J.: Completely regular designs of strength one. J. Algebr. Combin. 3, 177–185 (1994)

  7. 7.

    Mogilnykh, I.Y.: On the regularity of perfect 2-colorings of the Johnson graph. Prob. Inf. Transm. 43(4), 37–44 (2007)

  8. 8.

    Mogilnykh, I.Y., Avgustinovich, S.V.: Perfect 2-colorings of Johnson graphs \(J(6,3)\) and \(J(7,3)\). Lect. Notes Comput. Sci. 5228, 11–19 (2008)

Download references

Author information

Correspondence to Mehdi Alaeiyan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Behruz Tayfeh-Rezaie.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alaeiyan, M., Abedi, A. & Alaeiyan, M. Perfect 3-Colorings of the Johnson Graph J(6, 3). Bull. Iran. Math. Soc. (2020). https://doi.org/10.1007/s41980-019-00346-9

Download citation

Keywords

  • Perfect 3-colorings
  • Johnson graph
  • Breaking
  • Merging

Mathematics Subject Classification

  • 05C15
  • 68R05