On k-chromatically unique and k-chromatically equivalent hypergraphs

We consider a natural generalization of chromatically unique and chromatically equivalent notions in the class of hypergraphs.For a fixed λ,k∈N and a hypergraph H we denote by fk(H,λ) a number of different λ-colourings of H satisfying that an image of an edge e∈E(H) is at least a k-element set. It is the known fact that fk(H,λ) is a polynomial in λ [Drgas-Burchardt, E. and E. Łazuka, On chromatic polynomials of hypergraphs, Manuscript, (2006)].A hypergraph H is said to be k-chromatically unique if for each H1≠H we have fk(H1,λ)≠fk(H,λ). We call hypergraphs H1, H2 k-chromatically equivalent if fk(H1,λ)=fk(H,λ).In the paper we find a 3-chromatically unique class of hypergraphs. Moreover we use a given f3(H,λ) polynomial to characterize some class of hypergraphs.