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

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.