On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

Abstract: In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, and generalizes the coboundary polynomial of Crapo, and the bivariate chromatic polynomial of Dohmen, P\"onitz and Tittman. We show that specializations of these new polynomials recover polynomials which enumerate hyperedge coverings, matchings, transversals, and section hypergraphs. We also prove that the polynomials can be defined in terms of M\"obius inversion on the bond lattice of a hypergraph, as well as compute these polynomials for various classes of hypergraphs.