The large-N limit of gauge theories has been playing a crucial role in theoretical physics over the decades. Despite its importance, little is known outside the planar limit where the ’t Hooft coupling λ = g Y MN is fixed. In this Letter we consider more general large-N limit — λ grows with N , e.g., g YM is fixed. Such a limit is important particularly in recent attempts to find the nonpertuba...

|We continue the analysis of (r; q)-polycycles, i.e., planar graphs G that admit a realization on the plane such that all internal vertices have degree q, all boundary vertices have degree at most q, and all internal faces are combinatorial r-triangles; moreover, the vertices, edges, and internal faces form a cell complex. Two extremal problems related to chemistry are solved: the description o...

Split networks are a popular tool for the analysis and visualization of complex evolutionary histories. Every collection of splits (bipartitions) of a finite set can be represented by a split network. Here we characterize which collection of splits can be represented using a planar split network. Our main theorem links these collections of splits with oriented matroids and arrangements of lines...

In this paper, we show that every 4-connected maximal plane graph with m finite faces other than the octahedron can be drawn in the plane so that at least (m+3)/2 faces are acute triangles. Moreover, this bound is sharp. © 2005 Published by Elsevier B.V.

A rollercoaster is a sequence of real numbers for which every maximal contiguous subsequence, that is increasing or decreasing, has length at least three. By translating this sequence to a set of points in the plane, a rollercoaster can be defined as a polygonal path for which every maximal sub-path, with positiveor negative-slope edges, has at least three points. Given a sequence of distinct r...

A convex drawing of a plane graph G is a plane drawing of G, where each vertex is drawn as a point, each edge is drawn as a straight line segment and each face is drawn as a convex polygon. A maximal segment is a drawing of a maximal set of edges that form a straight line segment. A minimum-segment convex drawing of G is a convex drawing of G where the number of maximal segments is the minimum ...

,u.)-eUl''''''' and may be based on consideration of maximization of Lv~"HLVU'H"fJ chart scores or on the minimization of overall cost. We assume that all are pre;ter'able. Geometric duality that the adjacency graph be planar. context, this translates to maximal planarity, since adding further to

We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30n. This new bound is achieved by a careful optimization of the charging scheme of Sharir and Welzl (2006), which has led to the previous best upper bound of 43n for the problem. Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-fre...

A straight-line drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straight-line drawing is a maximal set of edges that form a straight line segment. A minimum-segment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all...

We consider a Delaunay triangulation defined on n points distributed independently and uniformly on a planar compact convex set of positive volume. Let the stabbing number be the maximal number of intersections between a line and edges of the triangulation. We show that the stabbing number Sn is Θ( √ n) in the mean, and provide tail bounds for P{Sn ≥ t √ n}. Applications to planar point locatio...