Duality pairs and homomorphisms to oriented and un-oriented cycles
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In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs (G,H) such that for any digraph, D, G→ D if and only if D↛H. The directed path on k+1 vertices together with the transitive tournament on k vertices is a classic example of a duality pair. This relation between paths and tournaments implies that a graph is k-colourable if and only if it admits an orientation with no directed path on more than k-vertices. In this work, for every undirected cycle C we find an orientation C_D and an oriented path P_C, such that (P_C,C_D) is a duality pair. As a consequence we obtain that there is a finite set, F_C, such that an undirected graph is homomorphic to C, if and only if it admits an F_C-free orientation. As a byproduct of the proposed duality pairs, we show that if T is a tree of height at most 3, one can choose a dual of T of linear size with respect to the size of T.
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