Convex circuit-free coloration of an oriented graph

We introduce the convex circuit-free coloration and convex circuit-free chromatic number −→ χa( −→ G) of an oriented graph −→ G and establish various basic results. We show that the problem of deciding if an oriented graph verifies χa( −→ G ) ≤ k is NP-complete if k ≥ 3 and polynomial if k ≤ 2. We exhibit an algorithm which finds the optimal convex circuit-free coloration for tournaments, and characterize the tournaments that are vertex-critical for the convex circuit-free coloration.