Perfect 2-colorings of Hamming graphs

Abstract

We consider the problem of existence of perfect 2-colorings of Hamming graphs with given parameters. We start with conditions on parameters of graphs and colorings that are necessary for their existence. Next we observe constructions of perfect colorings, including some new constructions giving new parameters of colorings. At last, we deduce which parameters of colorings are covered by these constructions and give tables of admissible parameters of 2-colorings in Hamming graphs H(n, q) for small n and q.

Minimum supports of eigenfunctions of graphs: a survey

“…The matrix S = (S i,j ) i,j∈{1,...,r} is called the quotient matrix of the equitable partition. A set C ⊆ V is called a 1-perfect code in G if every ball of radius 1 contains one vertex from C. For more information on equitable partitions and perfect codes we refer the reader to [9], [44,Chapter 5] and [1,47,88,89].…”