Perfect 2-colorings of the cubic graphs of order less than or equal to 10

Abstract

Perfect coloring is a generalization of the notion of completely regular codes, given by Delsarte. A perfect m-coloring of a graph G with m colors is a partition of the vertex set of G into m parts A1, . , Am such that, for all i,j∈{1,…,m}, every vertex of Ai is adjacent to the same number of vertices, namely, aij vertices, of Aj . The matrix A=(aij)i,j∈{1,…,m}, is called the parameter matrix. We study the perfect 2-colorings (also known as the equitable partitions into two parts) of the cubic graphs of order less than or equal to 10. In particular, we classify all the realizable parameter matrices of perfect 2-colorings for the cubic graphs of order less than or equal to 10.