Perfect colorings of hypergraphs

TL;DR

This paper develops foundational concepts for perfect colorings in hypergraphs, introduces a multidimensional matrix framework, and explores their properties, including eigenvalues, coverings, and specific examples like the Fano's plane hypergraph.

Contribution

It introduces a new multidimensional matrix approach for perfect hypergraph colorings and compares it with existing methods, advancing the theoretical understanding.

Findings

Eigenvalues of perfect colorings relate to hypergraph adjacency matrices.

Existence of common hypergraph coverings is established.

All perfect 2-colorings of Fano's plane are classified.

Abstract

Perfect colorings (equitable partitions) of graphs are extensively studied, while the same concept for hypergraphs attracts much less attention. The aim of this paper is to develop basic notions and properties of perfect colorings for hypergraphs. Firstly, we introduce a multidimensional matrix equation for perfect colorings of hypergraphs and compare this definition with a standard approach based on the incidence graph. Next, we show that the eigenvalues of the parameter matrix of a perfect coloring are eigenvalues of the multidimensional adjacency matrix of a hypergraph. We consider coverings of hypergraphs as a special case of perfect colorings and prove a theorem on the existence of a common covering of two hypergraphs. As an example, we show that a $k$-transversal in a hypergraph corresponds to a perfect coloring and calculate its parameters. At last, we find all perfect $2$-colorings of the Fano's plane hypergraph and compute some eigenvalues of this hypergraph.