On Oriented Arc-Coloring of Subcubic 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 (Recall that LD(G) is given by V (LD(G)) = A(G) and −→ ab ∈ A(LD(G)) whenever a = − →uv and b =−→ vw). We prove that every oriented subcubic graph has oriented chromatic index at most 7 and construct a subcubic graph with oriented chromatic index 6. AMS Subject Classification: 05C15.