3,448 research outputs found
A computational approach to Conway's thrackle conjecture
A drawing of a graph in the plane is called a thrackle if every pair of edges
meets precisely once, either at a common vertex or at a proper crossing. Let
t(n) denote the maximum number of edges that a thrackle of n vertices can have.
According to a 40 years old conjecture of Conway, t(n)=n for every n>2. For any
eps>0, we give an algorithm terminating in e^{O((1/eps^2)ln(1/eps))} steps to
decide whether t(n)2. Using this approach, we improve the
best known upper bound, t(n)<=3/2(n-1), due to Cairns and Nikolayevsky, to
167/117n<1.428n.Comment: 16 pages, 7 figure
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Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
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Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
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