  We perform extensive Monte Carlo simulations of the 10-state Potts model on
quenched two-dimensional $\Phi^3$ gravity graphs to study the effect of
quenched coordination number randomness on the nature of the phase transition,
which is strongly first order on regular lattices. The numerical data provides
strong evidence that, due to the quenched randomness, the discontinuous
first-order phase transition of the pure model is softened to a continuous
transition, representing presumably a new universality class. This result is in
striking contrast to a recent Monte Carlo study of the 8-state Potts model on
two-dimensional Poissonian random lattices of Voronoi/Delaunay type, where the
phase transition clearly stayed of first order, but is in qualitative agreement
with results for quenched bond randomness on regular lattices. A precedent for
such softening with connectivity disorder is known: in the 10-state Potts model
on annealed Phi3 gravity graphs a continuous transition is also observed.
