A Construction of Uniquely n-Colorable Digraphs with Arbitrarily Large Digirth

A natural digraph analogue of the graph-theoretic concept of an ‘independent set’ is that of an ‘acyclic set’, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets and we say a digraph is uniquely n-colorable when this decomposition is unique up to relabeling. It was shown probabilistically in [A. Harutyunyan et al., Uniquely D-colorable digraphs with large girth, Canad. J. Math., 64(6): 1310– 1328, 2012] that there exist uniquely n-colorable digraphs with arbitrarily large girth. Here we give a construction of such digraphs and prove that they have circular chromatic number n. The graph-theoretic notion of ‘homomorphism’ also gives rise to a digraph analogue. An acyclic homomorphism from a digraph D to a digraph H is a mapping φ : V (D) → V (H) such that uv ∈ A(D) implies that either φ(u)φ(v) ∈ A(H) or φ(u) = φ(v), and all the ‘fibers’ φ−1(v), for v ∈ V (H), of φ are acyclic. In this language, a core is a digraph D for which there does not exist an acyclic homomorphism from D to a proper subdigraph of itself. Here we prove some basic results about digraph cores and construct highly chromatic cores without short cycles.