  We establish variants of the Lefschetz hyperplane section theorem for the
integral tropical homology groups of tropical hypersurfaces of toric varieties.
It follows from these theorems that the integral tropical homology groups of
non-singular tropical hypersurfaces which are compact or contained in
$\mathbb{R}^n$ are torsion free. We prove a relationship between the
coefficients of the $\chi_y$ genera of complex hypersurfaces in toric varieties
and Euler characteristics of the integral tropical cellular chain complexes of
their tropical counterparts. It follows that the integral tropical homology
groups give the Hodge numbers of compact non-singular hypersurfaces of complex
toric varieties. Finally for tropical hypersurfaces in certain affine toric
varieties, we relate the ranks of their tropical homology groups to the
Hodge-Deligne numbers of their complex counterparts.
