  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.
