Acyclic and oriented chromatic numbers of graphs

The oriented chromatic number χo(~G) of an oriented graph ~G = (V,A) is the minimum number of vertices in an oriented graph ~H for which there exists a homomorphism of ~G to ~H. The oriented chromatic number χo(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relation between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for χo(G) in terms of χa(G). An upper bound for χo(G) in terms of χa(G) was given by Raspaud and Sopena. We also give an upper bound for χo(G) in terms of the maximum degree of G. This upper bound improves an earlier upper bound of the same kind. We shall show that this upper bound is not far from being optimal. 1