In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A k-stack (resp...

A queue layout of a graph consists of a total order of the vertices, and a partition of the edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph is its queue-number. A three-dimensional (straight- line grid) drawing of a graph represents the vertices by points in Z and the edges by non-crossing line-segments. This paper contributes three main results: (1) It is proved that the minimum volume of a certain type of three-dimensional drawing of a graph G is closely related to the queue-number of G. In particular, if G is an n-vertex member of a proper minor-closed family of graphs (such as a planar graph), then G has a O(1) O(1) O(n) drawing if and only if G has O(1) queue-number.

We prove the theorem from the title: the acyclic edge chromatic number of a random d-regular graph is asymptotically almost surely

. 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

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded non-repetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologically-closed class, while graphs with bounded crossing number are contained in a minor-closed class.

We relate acyclic (and star) chromatic number of a graph to the chromatic number of its minors and as a consequence we show that the set of all triangle free planar graphs is is homomorphism bounded by a triangle free graph. This solves a problem posed in [14] and completes results of [13] where a similar result has been proved for Kk -free graphs for k = 4; 5. It also

A vertex coloring of a graph is acyclic if the ends of each edge have different colors and each bicolored subgraph is acyclic. Let each face of size at most k, where k , in a map on a surface S N be replaced by a clique on the same vertices. Then the resulting map can be acyclically colored with a number of colors that depends linearly both on N and on k. Such results were previously known only for 1 N 2 and 3 k 4. 1 Introduction The problems of coloring graphs embedded on surfaces is an important part of graph theory. In particular, the well-known Four Color Problem belongs to this area. We denote by V (G) the set of vertices of a graph G and by E(G) its set of edges. A (proper) k-coloring of G is a mapping f : V (G) \Gamma! f1; 2; : : : ; kg such that f(x) 6= f(y) whenever x and y are adjacent in G. A part of this research was done during a visit of LaBRI, supported by the NATO Collaborative Research Grant n o 97-1519. It was partially supported by the grant 97-01-01075...

We de ne folding of a directed graph as a coloring (or a homomorphism) which is injective on all the down sets of a given depth. While in general foldings are as complicated as homomorphisms for some some classes they present an useful tool to study colorings and homomorphisms. Our main result yields for any proper minor closed class K a folding (of any prescribed depth) using a xed number of colors. This in turn yields (for any K) the existence of a K k -free graph which bounds all K k -free graphs belonging to K. This has been conjectured in [9] and elsewhere and solved for k = 3 in [10].

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

Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several linear-time algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of so-called topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with Erdös-Rényi model of random graphs with constant average degree. In particular, we prove that for every fixed d >