Acyclic and oriented chromatic numbers of graphs

The oriented chromatic number χ_o(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 χ_o(G) of an undirected graph G is the maximum of the oriented chromatic numbers of all the orientations of G. This paper discusses the relations between the oriented chromatic number and the acyclic chromatic number and some other parameters of a graph. We shall give a lower bound for χ_o(G) in terms of χ_a(G). An upper bound for χ_o(G) in terms of χ_a(G) was given by Raspaud and Sopena. We also give an upper bound for χ_o(G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. © 1997 John Wiley & Sons, Inc.