Homomorphisms to powers of digraphs Original Research Article

Given a digraph G and a sufficiently long directed path P, a folklore result says that G is homomorphic to P if and only if all cycles in G are balanced (the same number of forward and backward arcs). The purpose of this paper is to study homomorphisms of digraphs G that contain unbalanced cycles. In this case, we may be able to find a homomorphism of G to a power of P. Our main result states that the minimum power of P to which G admits a homomorphism equals the maximum imbalance (ratio of forward and backward arcs) of any cycle in G. The proof also yields a polynomial time algorithm to find this minimum power of P. We identify a larger class of digraphs H for which this minimum power problem can be solved in polynomial time, it includes all oriented paths H. By relating our powers of paths to complete graphs and so-called circular graphs, we are able to deduce a classical result of Minty regarding the chromatic number, as well as its more recent extension, by Goddyn, Tarsi, and Zhang, to the circular chromatic number.