Convex circuit-free coloration of an oriented graph

We introduce the Convex Circuit-Free coloration and Convex Circuit-Free chromatic number χ a ⃗ ( G ⃗ ) of an oriented graph G ⃗ and establish various basic results. We show that the problem of deciding if an oriented graph verifies χ a ( G ⃗ ) ≤ k is NP-complete if k ≥ 3 , and polynomial if k ≤ 2 . We introduce 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.