On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

TL;DR

This paper introduces multivariate hyperedge elimination and chromatic polynomials for hypergraphs, generalizing existing polynomials and connecting them to hypergraph combinatorial structures through deletion-contraction and M"obius inversion.

Contribution

It defines new multivariate polynomials for hypergraphs, generalizes previous polynomials, and links them to hypergraph enumeration problems and lattice theory.

Findings

Polynomials recover hyperedge coverings, matchings, and transversals.

Polynomials can be expressed via M"obius inversion on hypergraph bond lattices.

Explicit computations provided for various hypergraph classes.

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.