Abstract
In the homomorphism order of digraphs, a duality pair is an ordered pair of digraphs $(G,H)$ such that for any digraph, $D$, $G\to D$ if and only if $D\not \to H$. The directed path on $k+1$ vertices together with the transitive tournament on $k$ vertices is a classic example of a duality pair.
 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 an oriented tree of height at most $3$, one can choose a dual of $T$ of linear size with respect to the size of $T$.
Highlights
We consider graphs and digraphs with neither loops nor parallel arcs
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
For each positive integer n, n 3, we present a finite set of oriented graphs Fn such that Fn-graphs are precisely Cn-colourable graphs, i.e., graphs that admit a homomorphism to the n-cycle
Summary
We consider graphs and digraphs with neither loops nor parallel arcs. For a digraph D, we use VD to denote its vertex set and AD to denote its arc set. For any oriented tree, T , there is a digraph DT (the dual of T ), such that (T, DT ) is a duality pair in the homomorphism order of digraphs Their result is more general, dealing with relational structures, so, as other authors have done, we consider a restriction for the context of this work. In terms of F - graphs, the Roy-Gallai-Vitaver-Hasse Theorem states that, when F is the set of oriented graphs on k + 1 vertices with a hamiltonian directed path, the class of F -graphs is the class of k-colourable graphs. In this case, one can assume that such an orientations is acyclic.
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