A family of universal pseudo-homogeneous G-colourable graphs

For each finite core graph G there is a countable universal pseudo-homogeneous G-colourable graph M(G) that is unique up to isomorphism. We investigate properties of M(G) that are not unlike properties of the infinite random graph. In particular, we show that M(G) has an independent dominating set and has one- and two-way hamiltonian paths when G is connected. We also investigate limits of the graphs M(Gi), and we answer a question of Caicedo (Algebra Universalis 34 (1995) 314) on infinite antichains in the lattice of cores.