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 $[5]$.