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...

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.

Vertex rewriting in graphs is a very powerful mechanism which has been studied for quite a long time. In this paper we eventually provide a categorical theory of vertex rewriting and show how it can extend in a uniform way to node and pattern rewriting mechanisms in hypergraphs.

. Pullback rewriting has recently been introduced as a new and unifying paradigm for vertex rewriting in graphs. In this paper we show how to extend it to describe in a uniform way more rewriting mechanisms such as node and handle rewriting in hypergraphs. 1 Introduction After more than twenty years of (hyper)graph rewriting, a large number of rewriting mechanisms have been introduced which generate various classes of (hyper)graphs with different properties. A huge number of papers has been devoted to the classification and comparison of all those rewriting techniques, proposing various kind of encoding - sometimes fairly complicated - to help compare different classes of languages. Quoting them would significantly increase the size of this paper but we can refer the reader to at least [?, ?, ?, ?]. Still, since most of those works propose their own mechanism, their own formalism - in general an ad hoc set-theoretic one - this comparison is quite difficult and the necessity of a unif...

. The very extensive literature on rewriting - be it words, terms, graphs or whatever -, basically relies on the definition of a notion of substitution which can be intuitively understood as the succession of three basic operations : deletion of the part to be rewritten to provide a context, union of this context with the right hand side of a rule, liaison of the those two parts, most often by identification of some corresponding items. In the field of graph rewriting, this has led to the very elegant, very productive and therefore very popular method known as the double push-out approach to graph rewriting [19]. Yet this method has met its descriptive limits when trying to deal with the various notions of node replacement which have been introduced in [22] or [23]. In this paper we show how - when set in a proper framework - products (or pullbacks) can provide a very generic and uniform rewriting mechanism encompassing all known rewriting techniques and extending uniformly to arbitrar...