The complexity of colouring by semicomplete digraphs
SIAM Journal on Discrete Mathematics
On the complexity of H-coloring
Journal of Combinatorial Theory Series B
The effect of two cycles on the complexity of colouring by directed graphs
Discrete Applied Mathematics
Discrete Applied Mathematics
On (k, d)-colorings and fractional nowhere-zero flows
Journal of Graph Theory
Note: A connection between circular colorings and periodic schedules
Discrete Applied Mathematics
Minimum cost homomorphisms to reflexive digraphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Resource-sharing systems and hypergraph colorings
Journal of Combinatorial Optimization
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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.