Perfect 2-colorings of Hamming graphs

TL;DR

This paper investigates the existence and construction of perfect 2-colorings in Hamming graphs, establishing parameter conditions, introducing new constructions, and providing tables of admissible parameters, culminating in the creation of a specific orthogonal array.

Contribution

It offers new constructions of perfect 2-colorings in Hamming graphs and determines admissible parameters, expanding the understanding of equitable partitions in these graphs.

Findings

Derived necessary conditions for perfect 2-colorings.

Proposed new constructions for perfect colorings with specific parameters.

Generated tables of admissible parameters for small Hamming graphs.

Abstract

We consider the problem of existence of perfect $2$-colorings (equitable $2$-partitions) 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 known constructions of perfect colorings and propose some new ones giving new parameters. 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 $q$. Using the connection with perfect colorings, we construct an orthogonal array OA(2048,7,4,5).