On Partitions of Two-Dimensional Discrete Boxes

Let $A$ and $B$ be finite sets and consider a partition of the \emph{discrete box} $A \times B$ into \emph{sub-boxes} of the form $A' \times B'$ where $A' \subset A$ and $B' \subset B$. We say that such a partition has the $(k,\ell)$-piercing property for positive integers $k$ and $\ell$ if every \emph{line} of the form $\{a\} \times B$ intersects at least $k$ sub-boxes and every line of the form $A \times \{b\}$ intersects at least $\ell$ sub-boxes. We show that a partition of $A \times B$ that has the $(k, \ell)$-piercing property must consist of at least $(k-1)+(\ell-1)+\left\lceil 2\sqrt{(k-1)(\ell-1)} \right\rceil$ sub-boxes. This bound is nearly sharp (up to one additive unit) for every $k$ and $\ell$. As a corollary we get that the same bound holds for the minimum number of vertices of a graph whose edges can be colored red and blue such that every vertex is part of red $k$-clique and a blue $\ell$-clique.Comment: 10 pages, 2 figure

Coloring Points with Respect to Squares

We consider the problem of 2-coloring geometric hypergraphs. Specifically, we show that there is a constant m such that any finite set S of points in the plane can be 2-colored such that every axis-parallel square that contains at least m points from S contains points of both colors. Our proof is constructive, that is, it provides a polynomial-time algorithm for obtaining such a 2-coloring. By affine transformations this result immediately applies also when considering homothets of a fixed parallelogram

On the size of planarly connected crossing graphs

We prove that if an $n$-vertex graph $G$ can be drawn in the plane such that each pair of crossing edges is independent and there is a crossing-free edge that connects their endpoints, then $G$ has $O(n)$ edges. Graphs that admit such drawings are related to quasi-planar graphs and to maximal $1$-planar and fan-planar graphs.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

An Almost Optimal Bound on the Number of Intersections of Two Simple Polygons

What is the maximum number of intersections of the boundaries of a simple m-gon and a simple n-gon, assuming general position? This is a basic question in combinatorial geometry, and the answer is easy if at least one of m and n is even. If both m and n are odd, the best known construction has mn-(m+n)+3 intersections, and it is conjectured that this is the maximum. However, the best known upper bound is only mn-(m + ? n/6 ?), for m ? n. We prove a new upper bound of mn-(m+n)+C for some constant C, which is optimal apart from the value of C

The maximum size of adjacency-crossing graphs

An adjacency-crossing graph is a graph that can be drawn such that every two edges that cross the same edge share a common endpoint. We show that the number of edges in an $n$-vertex adjacency-crossing graph is at most $5n-10$. If we require the edges to be drawn as straight-line segments, then this upper bound becomes $5n-11$. Both of these bounds are tight. The former result also follows from a very recent and independent work of Cheong et al.\cite{cheong2023weakly} who showed that the maximum size of weakly and strongly fan-planar graphs coincide. By combining this result with the bound of Kaufmann and Ueckerdt\cite{KU22} on the size of strongly fan-planar graphs and results of Brandenburg\cite{Br20} by which the maximum size of adjacency-crossing graphs equals the maximum size of fan-crossing graphs which in turn equals the maximum size of weakly fan-planar graphs, one obtains the same bound on the size of adjacency-crossing graphs. However, the proof presented here is different, simpler and direct.Comment: 17 pages, 11 figure

A note on light geometric graphs

Let G be a geometric graph on n vertices in general position in the plane. We say that G is k-light if no edge e of G has the property that each of the two open half-planes bounded by the line through e contains more than k edges of G. We extend the previous result in Ackerman and Pinchasi (2012) [1] and with a shorter argument show that every k-light geometric graph on n vertices has at most O(n√k) edges. This bound is best possible.Simons Foundation (Fellowship)National Science Foundation (U.S.) (Grant DMS-1069197)NEC Corporation (MIT Award