  Barrett and Crane have proposed a model of simplicial Euclidean quantum
gravity in which a central role is played by a class of Spin(4) spin networks
called "relativistic spin networks" which satisfy a series of physically
motivated constraints. Here a proof is presented that demonstrates that the
intertwiner of a vertex of such a spin network is uniquely determined, up to
normalization, by the representations on the incident edges and the
constraints. Moreover, the constraints, which were formulated for four valent
spin networks only, are extended to networks of arbitrary valence, and the
generalized relativistic spin networks proposed by Yetter are shown to form the
entire solution set (mod normalization) of the extended constraints. Finally,
using the extended constraints, the Barrett-Crane model is generalized to
arbitrary polyhedral complexes (instead of just simplicial complexes)
representing spacetime. It is explained how this model, like the Barret-Crane
model can be derived from BF theory by restricting the sum over histories to
ones in which the left handed and right handed areas of any 2-surface are
equal. It is known that the solutions of classical Euclidean GR form a branch
of the stationary points of the BF action with respect to variations preserving
this condition.
