Eight times four bialgebras of hypergraphs, cointeractions, and chromatic polynomials

TL;DR

This paper develops a framework of eight interconnected bialgebras related to hypergraphs, revealing their algebraic structures, dualities, and associated chromatic polynomials, extending previous work on graph bialgebras.

Contribution

It introduces eight quartets of bialgebras of hypergraphs, linking them with chromatic polynomials and dualities, expanding the algebraic understanding of hypergraph invariants.

Findings

Hypergraph bialgebra has a cointeracting bialgebra structure.

Hypergraph chromatic polynomial is derived from the associated bialgebra.

Eight quartets of bialgebras and their related chromatic polynomials are constructed.

Abstract

We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a double bialgebra in the sense of L. Foissy, who recently proved there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial ring ${\mathbb Q}[x]$. We show the polynomial associated to a hypergraph is the hypergraph chromatic polynomial. Moreover hypergraphs occurs in quartets: there is a dual, a complement, and a dual complement hypergraph. These correspondences are involutions and give rise to three other double bialgebras, and three more chromatic polynomials. In all we give eight quartets of bialgebras which includes recent bialgebras of M. Aguiar and F. Ardila, and by L. Foissy.