Chromatic polynomials of hypergraphs

We consider a natural generalization of the chromatic polynomial of a graph. Let the symbol f(x1,…,xm)(H,λ)f(x1,…,xm)(H,λ) denote a number of different λλ-colourings of a hypergraph H=(X,E)H=(X,E), where X={v1,…,vn}X={v1,…,vn} and E={e1,…,em}E={e1,…,em}, satisfying that in an edge eiei there are used at least xixi different colours. In the work we show that f(x1,…,xm)(H,λ)f(x1,…,xm)(H,λ) can be expressed by a polynomial in λλ of degree nn and as a sum of graph chromatic polynomials. Moreover, we present a reduction formula for calculating f(x1,…,xm)(H,λ)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)f(x1,…,xm)(H,λ) and their connection with the sizes of the edges of HH.