34 research outputs found
Annular and pants thrackles
A thrackle is a drawing of a graph in which each pair of edges meets
precisely once. Conway's Thrackle Conjecture asserts that a thrackle drawing of
a graph on the plane cannot have more edges than vertices. We prove the
Conjecture for thrackle drawings all of whose vertices lie on the boundaries of
connected domains in the complement of the drawing. We also give a
detailed description of thrackle drawings corresponding to the cases when
(annular thrackles) and (pants thrackles).Comment: 17 page
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
Convex Hull Thrackles
A \emph{thrackle} is a graph drawn in the plane so that every pair of its
edges meet exactly once, either at a common end vertex or in a proper crossing.
Conway's thrackle conjecture states that the number of edges is at most the
number of vertices. It is known that this conjecture holds for linear
thrackles, i.e., when the edges are drawn as straight line segments.
We consider \emph{convex hull thrackles}, a recent generalization of linear
thrackles from segments to convex hulls of subsets of points. We prove that if
the points are in convex position then the number of convex hulls is at most
the number of vertices, but in general there is a construction with one more
convex hull. On the other hand, we prove that the number of convex hulls is
always at most twice the number of vertices
Thrackles: An improved upper bound
A thrackle is a graph drawn in the plane so that every pair of its edges meet exactly once: either at a common end vertex or in a proper crossing. We prove that any thrackle of n vertices has at most 1.3984n edges. Quasi-thrackles are defined similarly, except that every pair of edges that do not share a vertex are allowed to cross an odd number of times. It is also shown that the maximum number of edges of a quasi-thrackle on n vertices is 3/2(n-1), and that this bound is best possible for infinitely many values of n. (C) 2019 Published by Elsevier B.V
Density theorems for intersection graphs of t-monotone curves
A curve \gamma in the plane is t-monotone if its interior has at most t-1
vertical tangent points. A family of t-monotone curves F is \emph{simple} if
any two members intersect at most once. It is shown that if F is a simple
family of n t-monotone curves with at least \epsilon n^2 intersecting pairs
(disjoint pairs), then there exists two subfamilies F_1,F_2 \subset F of size
\delta n each, such that every curve in F_1 intersects (is disjoint to) every
curve in F_2, where \delta depends only on \epsilon. We apply these results to
find pairwise disjoint edges in simple topological graphs