  We solve a 4-(bond)-vertex model on an ensemble of 3-regular Phi3 planar
random graphs, which has the effect of coupling the vertex model to 2D quantum
gravity. The method of solution, by mapping onto an Ising model in field, is
inspired by the solution by Wu et.al. of the regular lattice equivalent -- a
symmetric 8-vertex model on the honeycomb lattice, and also applies to higher
valency bond vertex models on random graphs when the vertex weights depend only
on bond numbers and not cyclic ordering (the so-called symmetric vertex
models).
  The relations between the vertex weights and Ising model parameters in the
4-vertex model on Phi3 graphs turn out to be identical to those of the
honeycomb lattice model, as is the form of the equation of the Ising critical
locus for the vertex weights. A symmetry of the partition function under
transformations of the vertex weights, which is fundamental to the solution in
both cases, can be understood in the random graph case as a change of
integration variable in the matrix integral used to define the model.
  Finally, we note that vertex models, such as that discussed in this paper,
may have a role to play in the discretisation of Lorentzian metric quantum
gravity in two dimensions.
