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We investigate the complexity of generalizations of colourings (acycliccolourings, $(k,\ell)$-colourings, homomorphisms, and matrix partitions), forthe class of transitive digraphs. Even though transitive digraphs are nicelystructured, many problems are intractable, and their complexity turns out to bedifficult to classify. We present some motivational results and several openproblems.

Colourings, Homomorphisms, and Partitions of Transitive Digraphs