On k-chromatically unique and k-chromatically equivalent hypergraphs

Abstract

Abstract 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 f k ( 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 f k ( 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 H 1 ≠ H we have f k ( H 1 , λ ) ≠ f k ( H , λ ) . We call hypergraphs H 1 , H 2 k-chromatically equivalent if f k ( H 1 , λ ) = f k ( H , λ ) . In the paper we find a 3-chromatically unique class of hypergraphs. Moreover we use a given f 3 ( H , λ ) polynomial to characterize some class of hypergraphs.