  The generalized Dehn-Sommerville relations determine the odd subalgebra of
the combinatorial Hopf algebra. We introduce a class of eulerian hypergraphs
that satisfy the generalized Dehn-Sommerville relations for the combinatorial
Hopf algebra of hypergraphs. We characterize a wide class of eulerian
hypergraphs according to the combinatorics of underlying clutters. The
analogous results hold for simplicial complexes by the isomorphism which is
induced from the correspondence of clutters and simplicial complexes.
