We consider the extension to directed graphs of the concepts of harmonious colouring and complete colouring. We give an upper bound for the harmonious chromatic number of a general directed graph, and show that determining the exact value of the harmonious chromatic number is NP-hard for directed graphs of bounded degree (in fact graphs with maximum indegree and outdegree 2); the complexity of the corresponding undirected case is not known. We also consider complete colourings, and show that in the directed case the existence of a complete colouring is NP-complete. We also show that the interpolation property for complete colourings fails in the directed case.
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