Chromatic polynomials of hypergraphs

Abstract

We consider a natural generalization of the chromatic polynomial of a graph. Let the symbol f ( x 1 , … , x m ) ( H , λ ) denote a number of different λ -colourings of a hypergraph H = ( X , E ) , where X = { v 1 , … , v n } and E = { e 1 , … , e m } , satisfying that in an edge e i there are used at least x i different colours. In the work we show that f ( x 1 , … , x m ) ( 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 ( x 1 , … , x m ) ( 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 ( x 1 , … , x m ) ( H , λ ) and their connection with the sizes of the edges of H .