Homomorphisms and colourings of oriented graphs: An updated survey

An oriented graph is a loopless digraph with no opposite arcs. An oriented kk-colouring of an oriented graph G⃗ is a partition of its set of vertices into kk parts in such a way that no two adjacent vertices belong to the same part, and all the arcs connecting every two parts have the same direction. Hence, such a colouring exists if and only if G⃗ admits a homomorphism to some oriented graph of order kk.The oriented chromatic number of G⃗ is then defined as the smallest kk for which G⃗ admits an oriented kk-colouring or, equivalently, as the minimum order of an oriented graph to which G⃗ admits a homomorphism.In this paper, we survey the main results about oriented colourings and propose a few open problems.