  A coloured graph is k-ultrahomogeneous if every isomorphism between two
induced subgraphs of order at most k extends to an automorphism. A coloured
graph is t-tuple regular if the number of vertices adjacent to every vertex in
a set S of order at most k depends only on the isomorphism type of the subgraph
induced by S.
  We classify the finite vertex-coloured k-ultrahomogeneous graphs and the
finite vertex-coloured l-tuple regular graphs for k at least 4 and l at least
5, respectively. Our theorem in particular classifies finite vertex-coloured
ultrahomogeneous graphs, where ultrahomogeneous means the graph is
simultaneously k-ultrahomogeneous for all k.
