Perfect 3-colorings of Cubic Graphs of Order $8$

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 $A_1$, $\dots$, $A_m$ such that, for all $ i,j\in \lbrace 1,\cdots ,m\rbrace $, every vertex of $A_i$ is adjacent to the same number of vertices, namely, $a_{ij}$ vertices, of $A_j$ . The matrix $A=(a_{ij})_{i,j\in \lbrace 1,\cdots ,m\rbrace }$ is called the parameter matrix. We study the perfect 3-colorings (also known as the equitable partitions into three parts) of the cubic graphs of order $8$. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the cubic graphs of order $8$.