57,121 research outputs found
Coxeter groups and random groups
For every dimension d, there is an infinite family of convex co-compact
reflection groups of isometries of hyperbolic d-space --- the superideal
(simplicial and cubical) reflection groups --- with the property that a random
group at any density less than a half (or in the few relators model) contains
quasiconvex subgroups commensurable with some member of the family, with
overwhelming probability.Comment: 18 pages, 14 figures; version 2 incorporates referee's correction
Covering Partial Cubes with Zones
A partial cube is a graph having an isometric embedding in a hypercube.
Partial cubes are characterized by a natural equivalence relation on the edges,
whose classes are called zones. The number of zones determines the minimal
dimension of a hypercube in which the graph can be embedded. We consider the
problem of covering the vertices of a partial cube with the minimum number of
zones. The problem admits several special cases, among which are the problem of
covering the cells of a line arrangement with a minimum number of lines, and
the problem of finding a minimum-size fibre in a bipartite poset. For several
such special cases, we give upper and lower bounds on the minimum size of a
covering by zones. We also consider the computational complexity of those
problems, and establish some hardness results
-covering red and blue points in the plane
We say that a finite set of red and blue points in the plane in general
position can be -covered if the set can be partitioned into subsets of
size , with points of one color and point of the other color, in
such a way that, if at each subset the fourth point is connected by
straight-line segments to the same-colored points, then the resulting set of
all segments has no crossings. We consider the following problem: Given a set
of red points and a set of blue points in the plane in general
position, how many points of can be -covered? and we prove
the following results:
(1) If and , for some non-negative integers and ,
then there are point sets , like -equitable sets (i.e.,
or ) and linearly separable sets, that can be -covered.
(2) If , and the points in are in convex position,
then at least points can be -covered, and this bound is tight.
(3) There are arbitrarily large point sets in general position,
with , such that at most points can be -covered.
(4) If , then at least points of
can be -covered. For , there are too many red points and at
least of them will remain uncovered in any -covering.
Furthermore, in all the cases we provide efficient algorithms to compute the
corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl
On the Parikh-de-Bruijn grid
We introduce the Parikh-de-Bruijn grid, a graph whose vertices are
fixed-order Parikh vectors, and whose edges are given by a simple shift
operation. This graph gives structural insight into the nature of sets of
Parikh vectors as well as that of the Parikh set of a given string. We show its
utility by proving some results on Parikh-de-Bruijn strings, the abelian analog
of de-Bruijn sequences.Comment: 18 pages, 3 figures, 1 tabl
Finding tight Hamilton cycles in random hypergraphs faster
In an -uniform hypergraph on vertices a tight Hamilton cycle consists
of edges such that there exists a cyclic ordering of the vertices where the
edges correspond to consecutive segments of vertices. We provide a first
deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton
cycles in random -uniform hypergraphs with edge probability at least . Our result partially answers a question of Dudek and Frieze [Random
Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton
cycles exists already for for and for
using a second moment argument. Moreover our algorithm is superior to
previous results of Allen, B\"ottcher, Kohayakawa and Person [Random Structures
& Algorithms 46 (2015), 446-465] and Nenadov and \v{S}kori\'c
[arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a
randomised polynomial time algorithm working for edge probabilities , while the algorithm of Nenadov and \v{S}kori\'c is a
randomised quasipolynomial time algorithm working for edge probabilities .Comment: 17 page
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