Hopf algebraic structures on hypergraphs and multi-complexes

TL;DR

This paper develops Hopf algebraic structures on hypergraphs and multi-complexes using species formalism, leading to new polynomial invariants and algebraic proofs of properties related to chromatic polynomials.

Contribution

It introduces cointeracting bialgebras on hypergraphs and multi-complexes, linking them to chromatic polynomials and providing algebraic proofs of their properties.

Findings

Defined two cointeracting bialgebras on hypergraphs

Derived Hopf algebraic proofs for chromatic polynomial properties

Established a Hopf algebra structure on multi-complexes as a quotient

Abstract

Using the formalism of species and twisted objects, we introduce two structures of cointeracting bialgebras on hypergraphs, induced by two notions of induced sub-hypergraphs. We study the associated unique morphisms of cointeracting bialgebras from hypergraphs to the polynomial algebra in one indeterminate: in the first case, this gives the chromatic polynomial of a graph attached to the considered hypergraph. In the second case, we obtained Helgason's notion of chromatic polynomial of a hypergraph. We obtain Hopf-algebraic proves of results about the values of this chromatic polynomial in -1 or about its coefficients, with the help of the action of a monoid of characters. This allows to give multiplicity-free formulas for the antipodes of these objects, using various notions of acyclic orientations of hypergraphs. Mixing the two notions of induced sub-hypergraphs, we obtain a third Hopf algebra, firstly described by Aguiar and Ardila. We obtain negative results on the existence of a second coproduct making it a cointeracting bialgebra. Anyway, it is still possible to obtain a polynomial invariant from this structure, which is the chomatic polynomial described by Aval, Kharagbossian and Tanasa. We nally study Iovanov and Jaiung's Hopf algebra of multi-complexes, making it a cointeracting bialgebra which has for quotient one of the preceding cointeracting bialgebras of hypergraphs.