Perfect 2-Colorings of Johnson Graphs J(6,3) and J(7,3)

The problem of the existence of perfect 2-colorings in Johnson graphs J(6, 3) and J(7, 3) is solved in this paper. Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte [3]. This problem of existence of such structures is closely related to Delsarte hypothesis about the nonexistence of nontrivial perfect codes in Johnson graphs, the problem of existence of block schemes, the problem of existence of completely regular codes in Johnson graphs and other well-known mathematical problems. Some auxiliary theorems, which can be applied for treatment of perfect colorings in two colors in other graphs, are given in this paper.