Perfect 3-colorings of Cubic Graphs of Order 8 - Alaeiyan, Mehdi; Mehrabani, Ayoob (Armenian Journal of Mathematics, 2018)
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, …, A_m such that, for all i,j1,… ,m brace , 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,j1,… ,m brace 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.