We prove that the minimum number of convex quadrilaterals determined by n points in general position in the plane – or in other words, the rectilinear crossing number of the complete graph Kn – is at… (More)

We introduce the <i>adaptive neighborhood graph</i> as a data structure for modeling a smooth manifold <i>M</i> embedded in some (potentially very high-dimensional) Euclidean space… (More)

2006 47th Annual IEEE Symposium on Foundations of…

2006

Up to the factor of 2, the result generalizes McMullen's upper bound theorem for convex polytopes (the case lscr = 0) and extends a theorem of Linhart for the case d les 4. Moreover, the bound… (More)

Let EMBEDk→d be the following algorithmic problem: Given a finite simplicial complex K of dimension at most k, does there exist a (piecewise linear) embedding of K into Rd? Known results easily imply… (More)

Given topological spaces <i>X</i>,<i>Y</i>, a fundamental problem of algebraic topology is understanding the structure of all continuous maps <i>X</i> → <i>Y</i>. We consider a computational version,… (More)

Eigenvalues associated to graphs are a well-studied subject. In particular the spectra of the adjacency matrix and of the Laplacian of random graphs G(n,p) are known quite precisely. We consider… (More)

Intersection graphs of disks and of line segments, respectively, have been well studied, because of both practical applications and theoretically interesting properties of these graphs. Despite… (More)

We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has… (More)