. 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...

. The oriented chromatic number o(H) of an oriented graph H is dened to be the minimum order of an oriented graph H 0 such that H has a homomorphism to H 0 . If each graph in a class K has a homomorphism to the same H 0 , then H 0 is K-universal. Let P k denote the class of orientations of planar graphs with girth at least k. Clearly, P 3 P 4 P 5 . . . We discuss the existence of P k -universal graphs with special properties. It is known (see [10]) that there exists a P 3 -universal graph on 80 vertices. We prove here that (1) there exist no planar P 4 -universal graphs; (2) there exists a planar P 16 -universal graph on 6 vertices; (3) for any k, there exist no planar P k -universal graphs of girth at least 6; (4) for any k, there exists a P 40k -universal graph of girth at least k + 1. Keywords. Oriented colorings, Planar graphs, Planar graphs with large girth, Universal graphs. 1

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.

. A graph is 1-planar if it can be drawn on the plane in such a way that every edge crosses at most one other edge. We prove that the acyclic chromatic number of every 1-planar graph is at most 20. Keywords. Acyclic colouring, planar graphs, 1-planar graphs. 1

Abstract not found

. The acyclic list chromatic number of every 1-planar graph is proved to be at most 7 and is conjectured to be at most 5. Keywords. Acyclic coloring, List coloring, Acyclic list coloring. 1

In this paper, we introduce the new notion of acyclic improper colorings of graphs. An improper coloring of a graph G is a mapping c from the set of vertices of G to a set of colors such that for every color i, the subgraph induced by the vertices with color i satisfies some property depending on i. Such an improper coloring is acyclic if for every two distinct colors i and j, the subgraph induced by all the edges linking a i-colored vertex and a j-colored vertex is acyclic. We prove that every outerplanar graph can be acyclically 2-colored in such a way that every monochromatic subgraph has degree at most five and that this result is best possible. For planar graphs, we prove some negative results and state some open problems. 1 Introduction Let G be a graph. We denote by V (G) the vertex set of G and by E(G) the edge set of G. A coloring of G is a mapping c from V (G) to a finite set of colors C. The mapping c is a k-coloring of G if the set C has k elements. A coloring c is pro...

Communicated by R. L. Grahm Abstract--A graph G is called (k, d)*-choosable if, for every list assignment L satisfying [L(v)l = k for all v E V(G), there is an L-coloring of G such that each vertex of G has at most d neighbors colored with the same color as itself. In this note, we prove that every planar graph without 4-cycles and

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-...