Oriented colorings of triangle-free planar graphs

Oriented graphs are directed graphs without opposite arcs. In other words an oriented graph is an orientation of an undirected graph, obtained by assigning to every edge one of the two possible orientations. IfG is a graph, V (G) denotes its vertex set, E(G) denotes its set of edges (or arcs if G is an oriented graph) and F (G) denotes its set of faces if G is planar. A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from V (G) to V (H) which preserves the arcs, that is (x, y) ∈ E(G) =⇒ (φ(x), φ(y)) ∈ E(H). We say that H is a target graph of G if there exists a homomorphism from G to H . The oriented chromatic number χo(G) of an oriented graph G is defined as the minimum order of a target graph of G. The oriented chromatic number χo(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. Finally, the oriented chromatic number χo(C) of a graph class C is defined as the maximum of χo(G) taken over all graphs G ∈ C. We will say that a graph G is H-colorable if H is a target graph of G and the vertices of H will be called colors.