Perfect 3-colorings of the cubic graphs of order 10 - Mehdi Alaeiyan; Ayoob Mehrabani (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2017)
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_2, ..., A_m such that, for all i,j 1, ... , 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 10. In particular, we classify all the realizable parameter matrices of perfect 3-colorings for the cubic graphs of order 10.