  We prove that in the Solovay model every OD graph G on reals satisfies one
and only one of the following two conditions:
  (I) G admits an OD colouring by ordinals;
  (II) there exists a continuous homomorphism of G_0 into G, where G_0 is a
certain F_sigma locally countable graph which is not R-OD colourable by
ordinals in the Solovay model.
  If the graph G is locally countable or acyclic then (II) can be strengthened
by the requirement that the homomorphism is a 1-1 map, i.e. an embedding.
  As the second main result we prove that Sigma^1_2 graphs admit the dichotomy
(I) vs. (II) in set--generic extensions of the constructible universe L
(although now (I) and (II) may be in general compatible). In this case (I) can
be strengthened to the existence of a Delta^1_3 colouring by countable ordinals
provided the graph is locally countable.
  The proofs are based on a topology generated by $\od$ sets.
