On Colouring Oriented Graphs of Large Girth

Abstract

We prove that for every oriented graph $D$ and every choice of positive integers $k$ and $\ell$, there exists an oriented graph $D^*$ along with a surjective homomorphism $\psi\colon D^* \to D$ such that: (i) girth$(D^*) \geq\ell$; (ii) for every oriented graph $C$ with at most $k$ vertices, there exists a homomorphism from $D^*$ to $C$ if and only if there exists a homomorphism from $D$ to C; and (iii) for every $D$-pointed oriented graph $C$ with at most $k$ vertices and for every homomorphism $\varphi\colon D^* \to C$ there exists a unique homomorphism $f\colon D \to C$ such that $\varphi=f \circ \psi$. Determining the chromatic number of an oriented graph $D$ is equivalent to finding the smallest integer $k$ such that $D$ admits a homomorphism to an order-$k$ tournament, so our main theorem yields results on the girth and chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given $\ell\geq 3$ and $k\geq 5$, we include a construction of an oriented graph with girth $\ell$ and chromatic number $k$.