. A digraph G is k-nice for some positive integer k if for every two (not necessarily distinct) vertices x and y in G and every pattern of length k, given as a sequence of pluses and minuses, there exists a walk of length k linking x to y which respects this pattern (pluses corresponding to forward edges and minuses to backward edges). A digraph is then nice if it is k-nice for some k. Similarly, a multigraph H , whose edges are coloured by a set of p colours, is k-nice if for every two (not necessarily distinct) vertices x and y in H and every pattern of length k, given as a sequence of colours, there exists a path of length k linking x to y which respects this pattern. Such a multigraph is nice if it is k-nice for some k. In this paper we study the structure of nice digraphs and multigraphs. Keywords. Graph homomorphisms, Oriented graphs, Edge-colored graphs, Universal graphs. 1

. 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...

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.

. For every positive integer k, we present an oriented graph G k such that deleting any vertex of G k decreases its oriented chromatic number by at least k and deleting any arc decreases the oriented chromatic number of G k by two. Keywords. Oriented colorings, Critical graphs. 1

. An oriented graph G is nice if there exists a positive integer k such that for every two vertices u; v (allowing u = v), and for every orientation of edges of the path of length k, there exists a walk of length k in G beginning at u and ending at v whose orientation of edges coincides with the given one. (Such a graph is also called k-nice.) We generalize this notion using the notion of a nilpotent semigroup of endomorphisms of (P(V ) + ; [), and consider two basic problems: (1) nd bounds for the nilpotency class of such semigroups in terms of their generators (in graph{theoretical language: provided that a graph G on n vertices is nice, nd the smallest k such that G is k-nice); (2) nd a way to demonstrate non-nilpotency of such semigroups (nd as simple as possible characterization of non-nice graphs). Keywords. Nice graphs, Nilpotent semigroup of endomorphisms. 1

Let C be a class of graphs. A graph H is C-universal if it admits a homomorphism from every graph in C. For certain classes of graphs{which we designate nitely constructible classes{we derive an algorithm to test a given graph H for C-universality. This algorithm also works in the category of (n; m)-mixed colored graphs, which include (for dierent values of m and n) oriented and edge coloured graphs. The algorithm

The (d; 1)-total number d (G) of a graph G is the width of the smallest range of integers that suffices to label the vertices and the edges of G so that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least d. This notion was introduced in [HY02]. In this paper, we prove that d (G) +2d 2 for connected planar graphs with large girth and high maximum degree . Our results are optimal for d = 2.