490 research outputs found
On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves
A long standing conjecture of Richter and Thomassen states that the total
number of intersection points between any simple closed Jordan curves in
the plane, so that any pair of them intersect and no three curves pass through
the same point, is at least .
We confirm the above conjecture in several important cases, including the
case (1) when all curves are convex, and (2) when the family of curves can be
partitioned into two equal classes such that each curve from the first class is
touching every curve from the second class. (Two curves are said to be touching
if they have precisely one point in common, at which they do not properly
cross.)
An important ingredient of our proofs is the following statement: Let be
a family of the graphs of continuous real functions defined on
, no three of which pass through the same point. If there are
pairs of touching curves in , then the number of crossing points is
.Comment: To appear in SODA 201
Nested cycles in large triangulations and crossing-critical graphs
We show that every sufficiently large plane triangulation has a large
collection of nested cycles that either are pairwise disjoint, or pairwise
intersect in exactly one vertex, or pairwise intersect in exactly two vertices.
We apply this result to show that for each fixed positive integer , there
are only finitely many -crossing-critical simple graphs of average degree at
least six. Combined with the recent constructions of crossing-critical graphs
given by Bokal, this settles the question of for which numbers there is
an infinite family of -crossing-critical simple graphs of average degree
Crossing-critical graphs with large maximum degree
A conjecture of Richter and Salazar about graphs that are critical for a
fixed crossing number is that they have bounded bandwidth. A weaker
well-known conjecture of Richter is that their maximum degree is bounded in
terms of . In this note we disprove these conjectures for every ,
by providing examples of -crossing-critical graphs with arbitrarily large
maximum degree
Counterexample to an extension of the Hanani-Tutte theorem on the surface of genus 4
We find a graph of genus and its drawing on the orientable surface of
genus with every pair of independent edges crossing an even number of
times. This shows that the strong Hanani-Tutte theorem cannot be extended to
the orientable surface of genus . As a base step in the construction we use
a counterexample to an extension of the unified Hanani-Tutte theorem on the
torus.Comment: 12 pages, 4 figures; minor revision, new section on open problem
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