Abstract. We investigate the maximum number of simple cycles and the maximum number of Hamiltonian cycles in a planar graph G with n vertices. Using the transfer matrix method we construct a family of graphs which have at least 2.4262 n simple cycles and at least 2.0845 n Hamilton cycles. Based on counting arguments for perfect matchings we prove that 2.3404 n is an upper bound for the number of Hamiltonian cycles. Moreover, we obtain upper bounds for the number of simple cycles of a given length with a face coloring technique. Combining both, we show that there is no planar graph with more than 2.8927 n simple cycles. This reduces the previous gap between the upper and lower bound for the exponential growth from 1.03 to 0.46.

A constant-work-space algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the input size. We show how to triangulate a plane straight-line graph with n vertices in O(n 2 ) time and constant workspace. We also consider the problem of preprocessing a simple polygon P for shortest path queries, where P is given by the ordered sequence of its n vertices. For this, we relax the space constraint to allow s words of work-space. After quadratic preprocessing, the shortest path between any two points inside P can be found in O(n 2 /s) time.

We obtain new lower and upper bounds for the maximum multiplicity of some weighted and, respectively, nonweighted common geometric graphs drawn on n points in the plane in general position (with no three points collinear): perfect matchings, spanning trees, spanning cycles (tours), and triangulations. (i) We present a new lower bound construction for the maximum number of triangulations a set of n points in general position can have. In particular, we show that a generalized double chain formed by two almost convex chains admits Ω(8.65 n ) different triangulations. This improves the bound Ω(8.48 n ) achieved by the previous best construction, the double zig-zag chain studied by Aichholzer et al. (ii) We obtain a new lower bound of Ω(12.00 n ) for the number of noncrossing spanning trees of the double chain composed of two convex chains. The previous bound, Ω(10.42 n ), stood unchanged for more than 10 years. (iii) Using a recent upper bound of 30 n for the number of triangulations, due to Sharir and Sheffer, we show that n points in the plane in general position admit at most O(68.62 n ) noncrossing spanning cycles. (iv) We derive lower bounds for the number of maximum and minimum weighted geometric graphs (matchings, spanning trees, and tours). We show that the number of shortest tours can be exponential in n for points in general position. These tours are automatically noncrossing. Likewise, we show that the number of longest noncrossing tours can be exponential in n. It was known that the number of shortest noncrossing perfect matchings can be exponential in n, and here we show that the number of longest noncrossing perfect matchings can be also exponential in n. It was known that the number of longest noncrossing spanning trees of a point set can be exponentially large, and here we show that this can be also realized with points in convex position. For points in convex position we re-derive tight bounds for the number of longest and shortest tours with some simpler arguments. We also give a combinatorial characterization of longest tours, which yields an O(n log n) time algorithm for computing them.

We prove that any planar graph on n vertices has less than O(5.2852 n ) spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less than O(2.7156 n ). As a consequence of the latter the grid size needed to realize a 3d polytope with integer coordinates can be bounded by O(147.7 n ). Our observations imply improved upper bounds for related quantities: the number of cycle-free graphs in a planar graph is bounded by O(6.4884 n ), the number of plane spanning trees on a set of n points in the plane is bounded by O(158.6 n ), and the number of plane cycle-free graphs on a set of n points in the plane is bounded by O(194.7 n ).

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In the Boundary Labeling problem, we are given a set of n points, referred to as sites, inside an axis-parallel rectangle R, and a set of n pairwise disjoint rectangular labels that are attached to R from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend.In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of R (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossingfree leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.

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A crossing-free straight-line drawing of a graph is monotone if there is a monotone path between any pair of vertices with respect to some direction. We show how to construct a monotone drawing of a tree with n vertices on an O(n 1.5 ) × O(n 1.5 ) grid whose angles are close to the best possible angular resolution. Our drawings are convex, that is, if every edge to a leaf is substituted by a ray, the (unbounded) faces form convex regions. It is known that convex drawings are monotone and, in the case of trees, also crossing-free.A monotone drawing is strongly monotone if, for every pair of vertices, the direction that witnesses the monotonicity comes from the vector that connects the two vertices. We show that every tree admits a strongly monotone drawing. For biconnected outerplanar graphs, this is easy to see. On the other hand, we present a simply-connected graph that does not have a strongly monotone drawing in any embedding.