Perfect 3-colorings on 6-regular graphs of order 9

Abstract

The concept of a perfect coloring, introduced by P. Delsarte, generalizes the concept of completely regular code. We study the perfect 3-colorings (also known as the equitable partitions into three parts) on 6-regular graphs of order 9. A perfect n-colorings of a graph is a partition of its vertex set. It splits vertices into n parts {{\displaystyle A}}_{\mathrm{1}},{{\displaystyle A}}_{\mathrm{2}},\mathrm{...},{{\displaystyle A}}_n such that for all i,j\in \{\mathrm{1},\mathrm{2},\mathrm{...},n\}, each vertex of A_i is adjacent to a_ij vertices of A_j. The matrix A={{\displaystyle ({{\displaystyle a}}_{ij})}}_{n\times n} is called quotient matrix or parameter matrix. In this article, we start by giving an algorithm to find all different types of 6-regular graphs of order 9. Then, we classify all the realizable parameter matrices of perfect 3-colorings on 6-regular graphs of order 9.

Keywords

Perfect coloring / equitable partition / regular graph

References

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