. The oriented chromatic number 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 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 o (G) in terms of a (G). An upper bound for o (G) in terms of a (G) was given by Raspaud and Sopena. We also give an upper bound for o (G) in terms of the maximum degree of G. We shall show that this upper bound is not far from being optimal. Keywords. Oriented chromatic number, Acyclic chromatic number. 1

The oriented chromatic number o(H) of an oriented graph H is defined as the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . The oriented chromatic number o(G) of an undirected graph G is then defined as the maximum oriented chromatic number of its orientations. In this paper we study the links between o(G) and mad(G) defined as the maximum average degree of the subgraphs of G. 1 Introduction and statement of results For every graph G we denote by V (G), with vG = jV (G)j, its set of vertices and by E(G), with e G = jE(G)j, its set of arcs or edges. A homomorphism from a graph G to a graph On leave of absence from the Institute of Mathematics, Novosibirsk, 630090, Russia. With support from Engineering and Physical Sciences Research Council, UK, grant GR/K00561, and from the International Science Foundation, grant NQ4000. y This work was partially supported by the Network DIMANET of the European Union and by the grant 96-01-01614 of the Russian F...

We discuss the existence of homomorphisms of arbitrary digraphs to a fixed oriented cycle C. Our main result asserts that if the cycle C is unbalanced then a digraph G is homomorphic to C if and only if (1) every oriented path homomorphic to G is also homomorphic to C, and (2) the length of every cycle of G is a multiple of the length of C. This answers a conjecture from an earlier paper with H. Zhou, and generalizes a result proved there. We also show that this characterization does not hold for balanced cycles. We relate these results to work on the complexity of homomorphism problems.

We prove that for any integers m; k, there is an integer n0 such that if G is a graph of girth n0 then any partial k-tree homomorphic to G is also homomorphic to C2m+1 . As a corollary, every non-bipartite graph does not have bounded treewidth duality.

A homomorphism of an oriented graph G = (V; A) to an oriented graph G = (V ; A ) is a mapping ' from V to V such that '(u)'(v) is an arc in G whenever uv is an arc in G. A homomorphism of G to G is said to be T -preserving for some oriented graph T if for every connected subgraph H of G isomorphic to a subgraph of T , H is isomorphic to its homomorphic image in G . The T -preserving oriented chromatic number ~ T (G) of an oriented graph G is the minimum number of vertices in an oriented graph G such that there exists a T -preserving homomorphism of G to G . This paper discusses the existence of T -preserving homomorphisms of oriented graphs. We observe that only families of graphs with bounded degree can have bounded T -preserving oriented chromatic number when T has both in-degree and out-degree at least two. We then provide some sufficient conditions for families of oriented graphs for having bounded T -preserving oriented chromatic number when T is a directed path or a directed tree.