Perfect Colorings of Submatrix Hypergraphs

Abstract

A submatrix hypergraph \( G_{n\times m} \) is a hypergraph whose vertices are entries of an \( n\times m \) matrix and hyperedges are submatrices of order \( 2 \). In this paper, we consider perfect colorings of submatrix hypergraphs and study their parameters. We provide several constructions of perfect colorings of \( G_{n\times m} \) and prove that the incidence matrices of \( 2 \)-designs are perfect colorings of the submatrix hypergraph. Moreover, we describe all perfect 2-colorings of hypergraphs \( G_{2\times m} \) and \( G_{3\times m} \).