Perfect 3-Colorings of the Platonic Graph

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 $$A_1, \ldots,A_m$$ such that, for all $$i,j\in \lbrace 1,\ldots ,m\rbrace$$ , every vertex of $$A_i$$ is adjacent to the same number of vertices, namely $$a_{ij}$$ vertices, of $$A_j$$ . The matrix $$A=(a_{ij})_{i,j\in \lbrace 1,\ldots ,m\rbrace }$$ 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.