591 research outputs found
Note on the number of edges in families with linear union-complexity
We give a simple argument showing that the number of edges in the
intersection graph of a family of sets in the plane with a linear
union-complexity is . In particular, we prove for intersection graph of a family of
pseudo-discs, which improves a previous bound.Comment: background and related work is now more complete; presentation
improve
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
Coloring intersection graphs of arc-connected sets in the plane
A family of sets in the plane is simple if the intersection of its any
subfamily is arc-connected, and it is pierced by a line if the intersection
of its any member with is a nonempty segment. It is proved that the
intersection graphs of simple families of compact arc-connected sets in the
plane pierced by a common line have chromatic number bounded by a function of
their clique number.Comment: Minor changes + some additional references not included in the
journal versio
Topological obstructions for vertex numbers of Minkowski sums
We show that for polytopes P_1, P_2, ..., P_r \subset \R^d, each having n_i
\ge d+1 vertices, the Minkowski sum P_1 + P_2 + ... + P_r cannot achieve the
maximum of \prod_i n_i vertices if r \ge d. This complements a recent result of
Fukuda & Weibel (2006), who show that this is possible for up to d-1 summands.
The result is obtained by combining methods from discrete geometry (Gale
transforms) and topological combinatorics (van Kampen--type obstructions) as
developed in R\"{o}rig, Sanyal, and Ziegler (2007).Comment: 13 pages, 2 figures; Improved exposition and less typos.
Construction/example and remarks adde
Grothendieck inequalities for semidefinite programs with rank constraint
Grothendieck inequalities are fundamental inequalities which are frequently
used in many areas of mathematics and computer science. They can be interpreted
as upper bounds for the integrality gap between two optimization problems: a
difficult semidefinite program with rank-1 constraint and its easy semidefinite
relaxation where the rank constrained is dropped. For instance, the integrality
gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen
as a Grothendieck inequality. In this paper we consider Grothendieck
inequalities for ranks greater than 1 and we give two applications:
approximating ground states in the n-vector model in statistical mechanics and
XOR games in quantum information theory.Comment: 22 page
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
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