Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

Abstract. The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆-regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆-regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th

We study straight-line drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of non-planar graphs with few slopes are also considered. For example, interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree.

Consider the open problem: does every complete geometric graph K 2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K 2n has a partition into n plane spanning trees. We characterise all such partitions. Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees. Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning. We prove that every complete geometric graph Kn can be partitioned into at most n n/12 plane trees.

We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).

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.

A d-dimensional hypercube drawing of a graph represents the vertices by distinct points in {0, 1} d, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections. 1

We study straight-line drawings of non-planar graphs with few slopes. Interval graphs, co-comparability graphs and AT-free graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.

Abstract. A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every n-vertex graph with bounded degeneracy has a three-dimensional grid drawing with O(n 3/2) volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the z-direction. We prove that every dag has an upward three-dimensional grid drawing with O(n 3) volume, which is tight for the complete dag. The previous best upper bound was O(n 4). Our main result concerning upward drawings is that every c-colourable dag (c constant) has an upward three-dimensional grid drawing with O(n 2) volume. This result matches the bound in the undirected case, and improves the best known bound from O(n 3) for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges. 1.

Abstract. In stack and queue layouts the vertices of a graph are linearly ordered from left to right, where each edge corresponds to an item and the left and right end vertex of each edge represents the addition and removal of the item to the used data structure. A graph admitting a stack or queue layout is a stack or queue graph, respectively. Typical stack and queue layouts are rainbows and twists visualizing the LIFO and FIFO principles, respectively. However, in such visualizations, twists cause many crossings, which make the drawings incomprehensible. We introduce linear cylindric layouts as a visualization technique for queue and deque (double-ended queue) graphs. It provides new insights into the characteristics of these fundamental data structures and extends to the visualization of mixed layouts with stacks and queues. Our main result states that a graph is a deque graph if and only if it has a plane linear cylindric drawing. 1