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.