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 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 $G_{3\times m}.$ Illustr. 1, bibliogr. 12.