André Raspaud

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.

Planar graphs without cycles of length from 4 to 7 are 3-colorable

Planar graphs without cycles of length from 4 to 7 are proved to be 3-colorable. Moreover, it is proved that each proper 3-coloring of a face of length from 8 to 11 in a connected plane graph without cycles of length from 4 to 7 can be extended to a proper 3-coloring of the whole graph. This improves on the previous results on a long standing conjecture of Steinberg.

Recursive graphs with small-world scale-free properties.

We discuss a category of graphs, recursive clique trees, which have small-world and scale-free properties and allow a fine tuning of the clustering and the power-law exponent of their discrete degree distribution. We determine relevant characteristics of those graphs: the diameter, degree distribution, and clustering parameter. The graphs have also an interesting recursive property, and generalize recent constructions with fixed degree distributions.

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.

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

On the vertex-arboricity of planar graphs

The vertex-arboricity a(G) of a graph G is the minimum number of subsets into which the set of vertices of G can be partitioned so that each subset induces a forest. It is well-known that a(G)@?3 for any planar graph G. In this paper we prove that a(G)@?2 whenever G is planar and either G has no 4-cycles or any two triangles of G are at distance at least 3.

On Star Coloring of Graphs

In this paper, we deal with the notion of star coloring of graphs. A star coloring of an undirected graph G is a proper vertex coloring of G (i.e., no two neighbors are assigned the same color) such that any path of length 3 in G is not bicolored.We give the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs and 2-dimensional grids. We also study and give bounds for the star chromatic number of other families of graphs, such as hypercubes, tori, d-dimensional grids, graphs with bounded treewidth and planar graphs.

Acyclic 4-Choosability of Planar Graphs Without Cycles of Specific Lengths

A proper vertex coloring of a graph G = (V, E) is acyclic if G contains no bicolored cycle. A graph G is acyclically L-list colorable if for a given list assignment L = {L(v): v ∈ V}, there exists a proper acyclic coloring c of G such that c(v) ∈ L(v) for all v ∈ V. If G is acyclically L-list colorable for any list assignment with |L(v)| ≥ k for all v ∈ V, then G is acyclically k-choosable.