On the oriented chromatic index of oriented graphs

A homomorphism from an oriented graph G to an oriented graph H is a mapping φ from the set of vertices of G to the set of vertices of H such that −−−−−−→ φ(u)φ(v) is an arc in H whenever −→ uv is an arc in G. The oriented chromatic index of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism from the line digraph LD(G) of G to H (the line digraph LD(G) of G is given by V (LD(G)) = A(G) and −→ ab ∈ A(LD(G)) whenever a = −→ uv and b = −→ vw). We give upper bounds for the oriented chromatic index of graphs with bounded acyclic chromatic number, of planar graphs and of graphs with bounded degree. We also prove that the problem of deciding whether an oriented graph has oriented chromatic index at most k is polynomial time if k ≤ 3 and is NP-complete if k ≥ 4.