On \(3\)-Chromatically Unique and \(3\)-Chromatically Equivalent Hypergraphs

Abstract

We introduce notions of \(k\)-chromatic uniqueness and \(k\)-chromatic equivalence in the class of all Sperner hypergraphs. They generalize the chromatic uniqueness and equivalence defined in the class of all graphs \([10]\) and hypergraphs \([2, 4, 8]\). Using some known facts, concerning a \(k\)-chromatic polynomial of a hypergraph \([5]\), a set of hypergraphs whose elements are \(3\)-chromatically unique is indicated. A set of hypergraphs characterized by a described \(3\)-chromatic polynomial is also shown. The application of the investigated notions can be found in \([5]\).