Eric Sopena

Good and semi-strong colorings of oriented planar graphs

Abstract A k-coloring of an oriented graph G = (V,A) is an assignment c of one of the colors 1,2, ⋯, k to each vertex of the graph such that, for every arc (x,y) of G, c(x) ≠ c(y). The k-coloring is good if for every arc (x,y) of G there is no arc (z,t) ϵ A such that c(x) = c(t) and c(y) = c(z). A k-coloring is said to besemi-strong if for every vertex x of G, c(z) ≠ c(t) for any pair {z,t} of vertices of N-(x). We show that every oriented planar graph has a good coloring using at most 5 x 24 colors and that every oriented planar graph G = (V,A) with d-(x) ⩽ 3 for every x ϵ V has a good and semi-strong coloring using at most 4 x 5 x 24 colors.

The chromatic number of oriented graphs

We introduce in this paper the notion of the chromatic number of an oriented graph G (that is of an antisymmetric directed graph) defined as the minimum order of an oriented graph H such that G admits a homomorphism to H. We study the chromatic number of oriented k-trees and of oriented graphs with bounded degree. We show that there exist oriented k-trees with chromatic number at least 2k+1 - 1 and that every oriented k-tree has chromatic number at most (k + 1) × 2k. For 2-trees and 3-trees we decrease these upper bounds respectively to 7 and 16 and show that these new bounds are tight. As a particular case, we obtain that oriented outerplanar graphs have chromatic number at most 7 and that this bound is tight too. We then show that every oriented graph with maximum degree k has chromatic number at most (2k - 1) × 22k-2. For oriented graphs with maximum degree 2 we decrease this bound to 5 and show that this new bound is tight. For oriented graphs with maximum degree 3 we decrease this bound to 16 and conjecture that there exists no such connected graph with chromatic number greater than 7.

Acyclic and oriented chromatic numbers of graphs

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. c 1997 John Wiley & Sons, Inc.

On the maximum average degree and the oriented chromatic number of a graph

Abstract The oriented chromatic number o( H ) of an oriented graph H is defined as the minimum order of an oriented graph H ′ such that H has a homomorphism to H ′. 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.

Oriented graph coloring

Abstract An oriented k -coloring of an oriented graph G (that is a digraph with no cycle of length 2) is a partition of its vertex set into k subsets such that (i) no two adjacent vertices belong to the same subset and (ii) all the arcs between any two subsets have the same direction. We survey the main results that have been obtained on oriented graph colorings.

Acyclic colouring of 1-planar graphs ☆

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.

Acyclic List 7-Coloring of Planar Graphs

The acyclic list chromatic number of every planar graph is proved to be at most 7.

Colorings and girth of oriented planar graphs

Abstract Homomorphisms between graphs are studied as a generalization of colorings and of chromatic number. We investigate here homomorphisms from orientations of undirected planar graphs to graphs (not necessarily planar) containing as few digons as possible. We relate the existence of such homomorphisms to girth and it appears that these questions remain interesting even if we insist the girth of G is large, an assumption which makes the chromatic number easy to compute. In particular, we prove that every orientation of any large girth planar graph is Scolorable and classify those digraphs on 3, 4 and 5 vertices which color all large girth oriented planar graphs. 1. Introduction and statement of results Given graphs G = (V, E) and G’ = (V’, E’) a homomorphism from G to G’ is any mapping f : V -+ V’ satisfying LGYI E E*[f(x),f(v)l E E’. Here the brackets on both sides of the implication means the same thing: either an edge or an arc. The existence of a homomorphism from G to G’ will be denoted by G + G’. Homomorphisms are clearly related to the chromatic number of undirected graphs (an undirected graph G is k-colorable if and only if there exists a homomorphism from G to

Incidence coloring of k-degenerated graphs

Abstract We prove that the incidence coloring number of every k -degenerated graph G is at most Δ ( G )+2 k −1. For K 4 -minor free graphs ( k =2), we decrease this bound to Δ ( G )+2, which is tight. For planar graphs ( k =5), we decrease this bound to Δ ( G )+7.

Different local controls for graph relabeling systems

We are interested in models to encode and to prove decentralized and distributed computations on graphs. In this paper we define and compare six models of graph relabeling systems. These systems do not change the underlying structure of the graph on which they work, but only the labeling of its components (edges or vertices). Each relabeling step is fully determined by the knowledge of a fixed-size subgraph, the local context of the relabeled occurrence. The families studied are based on the relabeling of partial or induced subgraphs and we use two kinds of mechanisms to control the applicability of rules locally: a priority relation on the set of rules or a set of forbidden contexts associated with each rule. We show that these two basic (i.e., without local control) families of graph relabeling systems are distinct, but whenever we consider the local controls of the relabeling, the four families so obtained are equivalent.