Perfect 3-Colorings of the Platonic Graph

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 is an element of{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 is an element of{1,...,m} is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the Platonic graphs. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the Platonic graphs.