Perfect Colorings of Submatrix Hypergraphs

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 andstudy 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, wedescribe all perfect 2-colorings of hypergraphs \(G_{2\times m}\) and \(G_{3\times m}\) .