Eight Times Four Bialgebras of Hypergraphs, Cointeractions, and Chromatic Polynomials

Abstract

Abstract The bialgebra of hypergraphs, a generalization of W. Schmitt’s Hopf algebra of graphs [25], is shown to have a cointeracting bialgebra structure, giving a double bialgebra in the sense of L. Foissy, who has recently proven [15] that there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial ring ${\mathbb Q}[x]$. We show that the associated polynomial is the hypergraph chromatic polynomial. Moreover, hypergraphs occur 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 addition to these two quartets of bialgebras we give six more, including recent bialgebras of hypergraphs introduced by M. Aguiar and F. Ardila [1] and by L. Foissy [17].