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

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.

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Correspondence to Mehdi Alaeiyan.

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Communicated by Behruz Tayfeh-Rezaie.

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Alaeiyan, M., Abedi, A. & Alaeiyan, M. Perfect 3-Colorings of the Johnson Graph J(6, 3). Bull. Iran. Math. Soc. 46, 1603–1612 (2020). https://doi.org/10.1007/s41980-019-00346-9

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Keywords

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

Mathematics Subject Classification

  • 05C15
  • 68R05