Complete oriented colourings and the oriented achromatic number

In this paper, we initiate the study of complete colourings of oriented graphs and the new associated notion of the oriented achromatic number of oriented and undirected graphs. In particular, we prove that for every integers a and b with 2 ≤ a ≤ b, there exists an oriented graph −→ Ga,b with oriented chromatic number a and oriented achromatic number b. We also prove that adding a vertex, deleting a vertex or deleting an arc in an oriented graph may increase its oriented achromatic number by an arbitrarily large value, while adding an arc may increase its oriented achromatic number by at most 2. Finally, we consider the behaviour of the oriented chromatic and achromatic numbers with respect to elementary homomorphisms and show in particular that, in contrast to the undirected case, there is no interpolation homomorphism theorem for oriented graphs. Our notion of complete colouring of oriented graphs corresponds to the notion of complete homomorphisms of oriented graphs and, therefore, differs from the notion of complete colourings of directed graphs recently introduced by Edwards in [Harmonious chromatic number of directed graphs. Discrete Appl. Math. 161 (2013), 369–376.].