On chromatic polynomials of hypergraphs

We consider a natural generalization of the chromatic polynomial of a graph. Let f(x1,…,xm)(H,λ) denote a number of different λ-colourings of a hypergraph H=(X,E),X={v1,…,vn}, E={e1,…em}, satisfying that in an edge ei it is used at least xi different colours. In the paper we show that f(x1,…,xm)(H,λ) can be expressed by a polynomial in λ of degree n and as a sum of graph chromatic polynomials. Moreover, we present a reduction formula for calculating f(x1,…,xm)(H,λ). It generalizes the similar formulas observed by H. Whitney and R.P. Jones for standard colourings of graphs and hypergraphs respectively. We also study some coefficients of f(x1,…,xm)(H,λ) and their connection with the sizes of the edges of H.