362 research outputs found
On Searching for Small Kochen-Specker Vector Systems (extended version)
Kochen-Specker (KS) vector systems are sets of vectors in R^3 with the
property that it is impossible to assign 0s and 1s to the vectors in such a way
that no two orthogonal vectors are assigned 0 and no three mutually orthogonal
vectors are assigned 1. The existence of such sets forms the basis of the
Kochen-Specker and Free Will theorems. Currently, the smallest known KS vector
system contains 31 vectors. In this paper, we establish a lower bound of 18 on
the size of any KS vector system. This requires us to consider a mix of
graph-theoretic and topological embedding problems, which we investigate both
from theoretical and practical angles. We propose several algorithms to tackle
these problems and report on extensive experiments. At the time of writing, a
large gap remains between the best lower and upper bounds for the minimum size
of KS vector systems.Comment: 16 pages. Extended version of "On Searching for Small Kochen-Specker
Vector Systems" published in WG 201
On complexity of mutlidistance graph recognition in
Let be a set of positive numbers. A graph is called an
-embeddable graph in if the vertices of can be
positioned in so that the distance between endpoints of any edge
is an element of . We consider the computational problem of
recognizing -embeddable graphs in and classify all
finite sets by complexity of this problem in several natural
variations.Comment: 38 pages, 9 figures. Extended abstract published in EUROCOMB'17
proceedings in Electronic Notes in Discrete Mathematics
(http://www.sciencedirect.com/science/article/pii/S1571065317302354
On the Complexity of Embeddable Simplicial Complexes
This thesis addresses the question of the maximal number of -simplices for
a simplicial complex which is embeddable into for some .
A lower bound of ,
which might even be sharp, is given by the cyclic polytopes. To find an upper
bound for the case we look for forbidden subcomplexes. A generalization
of the theorem of van Kampen and Flores yields those. Then the problem can be
tackled with the methods of extremal hypergraph theory, which gives an upper
bound of .
We also consider whether these bounds can be improved by simple means.Comment: Diplom thesis, FU Berlin, 200
On links of vertices in simplicial -complexes embeddable in the euclidean -space
We consider -dimensional simplicial complexes which can be PL embedded in
the -dimensional euclidean space. In short, we show that in any such
complex, for any three vertices, the intersection of the link-complexes of the
vertices is linklessly embeddable in the -dimensional euclidean space.
These considerations lead us to a new upper bound on the total number of
-simplices in an embeddable complex in -space with vertices,
improving known upper bounds, for all . Moreover, the bound is also
true for the size of -complexes linklessly embeddable in the
-dimensional space
Inapproximability for metric embeddings into R^d
We consider the problem of computing the smallest possible distortion for
embedding of a given n-point metric space into R^d, where d is fixed (and
small). For d=1, it was known that approximating the minimum distortion with a
factor better than roughly n^(1/12) is NP-hard. From this result we derive
inapproximability with factor roughly n^(1/(22d-10)) for every fixed d\ge 2, by
a conceptually very simple reduction. However, the proof of correctness
involves a nontrivial result in geometric topology (whose current proof is
based on ideas due to Jussi Vaisala).
For d\ge 3, we obtain a stronger inapproximability result by a different
reduction: assuming P \ne NP, no polynomial-time algorithm can distinguish
between spaces embeddable in R^d with constant distortion from spaces requiring
distortion at least n^(c/d), for a constant c>0. The exponent c/d has the
correct order of magnitude, since every n-point metric space can be embedded in
R^d with distortion O(n^{2/d}\log^{3/2}n) and such an embedding can be
constructed in polynomial time by random projection.
For d=2, we give an example of a metric space that requires a large
distortion for embedding in R^2, while all not too large subspaces of it embed
almost isometrically
Universal point sets for planar three-tree
For every , we present a set of
points in the plane such that every planar 3-tree with vertices has a
straight-line embedding in the plane in which the vertices are mapped to a
subset of . This is the first subquadratic upper bound on the size of
universal point sets for planar 3-trees, as well as for the class of 2-trees
and serial parallel graphs.Comment: revisio
Framed -valent Graph Minor Theory II: Special Minors and New Examples
In the present paper, we proceed the study of framed -graph minor theory
initiated in ``Framed -valent Graph Minor Theory I. Intoduction. Planarity
Criterion '' and justify the planarity theorem for arbitrary framed 4-graphs;
besides, we prove analogous results for embeddability in the projective plane.Comment: 12 page
The Projective Planarity Question for Matroids of -Nets and Biased Graphs
A biased graph is a graph with a class of selected circles ("cycles",
"circuits"), called "balanced", such that no theta subgraph contains exactly
two balanced circles. A biased graph has two natural matroids, the frame
matroid and the lift matroid. A classical question in matroid theory is whether
a matroid can be embedded in a projective geometry. There is no known general
answer, but for matroids of biased graphs it is possible to give algebraic
criteria. Zaslavsky has previously given such criteria for embeddability of
biased-graphic matroids in Desarguesian projective spaces; in this paper we
establish criteria for the remaining case, that is, embeddability in an
arbitrary projective plane that is not necessarily Desarguesian.
The criteria depend on the embeddability of a quasigroup associated to the
graph into the additive or multiplicative loop of a ternary coordinate ring for
the plane. A 3-node biased graph is equivalent to an abstract partial 3-net;
thus, we have a new algebraic criterion for an abstract 3-net to be realized in
a non-Desarguesian projective plane. We work in terms of a special kind of
3-node biased graph called a biased expansion of a triangle. Our results apply
to all finite 3-node biased graphs because, as we prove, every such biased
graph is a subgraph of a finite biased expansion of a triangle. A biased
expansion of a triangle, in turn, is equivalent to an isostrophe class of
quasigroups, which is equivalent to a -net. Much is not known about
embedding a quasigroup into a ternary ring, so we do not say our criteria are
definitive. For instance, it is not even known whether there is a finite
quasigroup that cannot be embedded in any finite ternary ring. If there is,
then there is a finite rank-3 matroid (of the corresponding biased expansion)
that cannot be embedded in any finite projective plane---a presently unsolved
problem.Comment: There are 2 figures. formerly part of arXiv:1608.06021v
Embedding Four-directional Paths on Convex Point Sets
A directed path whose edges are assigned labels "up", "down", "right", or
"left" is called \emph{four-directional}, and \emph{three-directional} if at
most three out of the four labels are used. A \emph{direction-consistent
embedding} of an \mbox{-vertex} four-directional path on a set of
points in the plane is a straight-line drawing of where each vertex of
is mapped to a distinct point of and every edge points to the direction
specified by its label. We study planar direction-consistent embeddings of
three- and four-directional paths and provide a complete picture of the problem
for convex point sets.Comment: 11 pages, full conference version including all proof
Embedding products of graphs into Euclidean spaces
For any collection of graphs we find the minimal dimension d such that the
product of these graphs is embeddable into the d-dimensional Euclidean space.
In particular, we prove that the n-th powers of the Kuratowsky graphs are not
embeddable into the 2n-dimensional Euclidean space. This is a solution of a
problem of Menger from 1929. The idea of the proof is the reduction to a
problem from so-called Ramsey link theory: we show that any embedding of L into
the (2n-1)-dimensional sphere, where L is the join of n copies of a 4-point
set, has a pair of linked (n-1)-dimensional spheres.Comment: in English and in Russian, 5 pages, 2 figures. Minor improvement of
exposition, a reference to a popular-science introduction adde
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