Abstract

A homomorphism of an oriented graph G to an oriented graph H is a mapping ' from the vertex set of G to the vertex set of H such that '(u)'(v) is an arc in H whenever uv is an arc in G. The oriented chromatic number of an oriented graph G is the minimum number of vertices in an oriented graph H such that there exists a homomorphism of G to H . We prove in this short note that every oriented graph with maximum degree 3 has oriented chromatic number at most 11. 1 Introduction An oriented graph is a digraph having no opposite arcs. Let G and H be two oriented graphs. A homomorphism of G to H is a mapping ' from the vertex set of G to the vertex set of H such that '(u)'(v) is an arc in H whenever uv is an arc in G. If G has a homomorphism to H we say that G is H-colorable. Such a homomorphism is called a H-coloring of G and the vertices of H are called colors. The oriented chromatic number of an oriented graph G is the minimum number of vertices in an oriented graph H such that G is H...