Crossing Numbers of Graphs is the first book devoted to the crossing number, an
increasingly popular object of study with surprising connections. The field has matured into a …

Among the intuitively appealing aspects of graph theory is its close connection to drawings
and geometry. The development of computer technology has become a source of motivation …

A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and
to each edge a simple continuous arc connecting the corresponding two points. The …

This chapter surveys computational topology results in the special, low-dimensional case
where the ambient space is a surface. Surface topology is very well-understood and …

In this paper, we present several density-type theorems which show how to find a copy of a
sparse bipartite graph in a graph of positive density. Our results imply several new bounds …

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A drawing of a graph is x-monotone if every edge intersects every vertical line at most once
and every vertical line contains at most one vertex. Pach and Tóth showed that if a graph …

An edge in a drawing of a graph is called even if it intersects every other edge of the graph
an even number of times. Pach and Tóth proved that a graph can always be redrawn so that …

A topological graph is a graph drawn in the plane so that its vertices are represented by
points, and its edges are represented by Jordan curves connecting the corresponding …

" Based on a lecture series given by the authors at a satellite meeting of the 2006
International Congress of Mathematicians and on many articles written by them and their …