Note: Professor Pach's number: [045]; Also in: Facsimile edition: World Classics in Mathematics Series, vol. 28, China Science Press, Beijing, 2006. Mimeographed from 100 Research Problems in Discrete Geometry (with W. O. J. Moser). Reference DCG-BOOK-2008-001 URL: http://www.math.nyu.edu/~pach/research_problems.html Record created on 2008-11-18, modified on 2017-05-12

Research Problems in Discrete Geometry

* Preliminaries* Numbers* Sequences and series* Limits and continuity* Differentiation* The elementary transcendental functions* Integration* Infinite series and infinite products* Trigonometric series* Bibliography* Other works by the author* Index

An introduction to classical real analysis, by Karl R. Stromberg. Pp 575. $29·95. 1981. ISBN 0-534-98012-0 (Wadsworth)

Graph Drawing Beyond Planarity is a rapidly growing research area that classifies and studies geometric representations of nonplanar graphs in terms of forbidden crossing configurations. The aim of this survey is to describe the main research directions in this area, the most prominent known results, and some of the most challenging open problems.

pdf/a-survey-on-graph-drawing-beyond-planarity-590emahil6.pdf

A Survey on Graph Drawing Beyond Planarity

An introduction to the theory of groups

The notion of 1-planarity is among the most natural and most studied generalizations of graph planarity. A graph is 1-planar if it has an embedding where each edge is crossed by at most another edge. The study of 1-planar graphs dates back to more than fifty years ago and, recently, it has driven increasing attention in the areas of graph theory, graph algorithms, graph drawing, and computational geometry. This annotated bibliography aims to provide a guiding reference to researchers who want to have an overview of the large body of literature about 1-planar graphs. It reviews the current literature covering various research streams about 1-planarity, such as characterization and recognition, combinatorial properties, and geometric representations. As an additional contribution, we offer a list of open problems on 1-planar graphs.

An annotated bibliography on 1-planarity

(1998). A Stroll Through the Gaussian Primes. The American Mathematical Monthly: Vol. 105, No. 4, pp. 327-337.

/pdf/a-stroll-through-the-gaussian-primes-15y1lwufaw.pdf

A Stroll Through the Gaussian Primes

A drawing of a graph G in the plane is said to be a rectilinear drawing of G if the edges are required to be line segments (as opposed to Jordan curves) We assume no three vertices are collinear The rectilinear crossing number of G is the fewest number of edge crossings attainable over all rectilinear drawings of G Thanks to Richard Guy, exact values of the rectilinear crossing number of K_n, the complete graph on n vertices, for n = 3,,9, are known (Guy 1972, White and Beineke 1978) Since 1971, thanks to the work of David Singer, the rectilinear crossing number of K_10 has been known to be either 61 or 62, a deceptively innocent and tantalizing statement The difficulty of determining the correct value is evidenced by the fact that Singer''s result has withstood the test of time In this paper we use a purely combinatorial argument to show that the rectilinear crossing number of K_10 is 62 Moreover, using this result, we improve an asymptotic lower bound for a related problem Finally, we close with some new and old open questions that were provoked, in part, by the results of this paper, and by the tangled history of the problem itself

/pdf/the-rectilinear-crossing-number-of-k-10-is-62-2ys6iviz6l.pdf

The Rectilinear Crossing Number of K_10 is 62

A drawing of a graph G in the plane is said to be a rectilinear drawing of G if the edges are required to be line segments (as opposed to Jordan curves). We assume no three vertices are collinear. The rectilinear crossing number of G is the fewest number of edge crossings attainable over all rectilinear drawings of G. Thanks to Richard Guy, exact values of the rectilinear crossing number of K_n, the complete graph on n vertices, for n = 3,...,9, are known (Guy 1972, White and Beinke 1978, Finch 2000, Sloanes A014540). Since 1971, thanks to the work of David Singer (1971, Gardiner 1986), the rectilinear crossing number of K_10 has been known to be either 61 or 62, a deceptively innocent and tantalizing statement. The difficulty of determining the correct value is evidenced by the fact that Singer's result has withstood the test of time. In this paper we use a purely combinatorial argument to show that the rectilinear crossing number of K_10 is 62. Moreover, using this result, we improve an asymptotic lower bound for a related problem. Finally, we close with some new and old open questions that were provoked, in part, by the results of this paper, and by the tangled history of the problem itself.

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S 1 such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are allowed to see through each other, the graphs representable in this way are precisely the interval graphs. We consider representations in which bars are allowed to see through at most k other bars. Since all bar visibility graphs are planar, we seek measurements of closeness to planarity for bar k-visibility graphs. We obtain an upper bound on the number of edges in a bar k-visibility graph. As a consequence, we obtain an upper bound of 12 on the chromatic number of bar 1-visibility graphs, and a tight upper bound of 8 on the size of the largest complete bar 1-visibility graph. We conjecture that bar 1-visibility graphs have thickness at most 2.

Bar k-visibility graphs : Bounds on the number of edges, chromatic number, and thickness

A triangulated polygon is a 2-connected maximal outerplanar graph A unit bar-visibility graph (UBVG for short) is a graph whose vertices can be represented by disjoint, horizontal, unit-length bars in the plane so that two vertices are adjacent if and only if there is a non-degenerate, unobstructed, vertical band of visibility between the corresponding bars We give combinatorial and geometric characterizations of the triangulated polygons that are UBVGs To each triangulated polygon G we assign a character string with the property that G is a UBVG if and only if the string satisfies a certain regular expression Given a string that satisfies this condition, we describe a linear-time algorithm that uses it to produce a UBV layout of G

/pdf/unit-bar-visibility-layouts-of-triangulated-polygons-ja7nvkfs1x.pdf

Unit bar-visibility layouts of triangulated polygons

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Papers

A Stroll Through the Gaussian Primes

The Rectilinear Crossing Number of K_10 is 62

The Rectilinear Crossing Number of K_10 is 62

Bar k-visibility graphs : Bounds on the number of edges, chromatic number, and thickness

Unit bar-visibility layouts of triangulated polygons