  I study the corrections engendered by the insertion of a "melon" graph in the
bulk of the first-order spinfoam used for the graviton propagator. I find that
these corrections are highly non-trivial and, in particular, that they concern
those terms which disappear in the Bojowald-Bianchi-Magliaro-Perini limit of
vanishing Barbero-Immirzi parameter at fixed area. This fact is the first
realization of the often cited idea that the spinfoam amplitude receives higher
order corrections under the refinement of the underlying two-complex.
