664 research outputs found
Triangle-free intersection graphs of line segments with large chromatic number
In the 1970s, Erdos asked whether the chromatic number of intersection graphs
of line segments in the plane is bounded by a function of their clique number.
We show the answer is no. Specifically, for each positive integer , we
construct a triangle-free family of line segments in the plane with chromatic
number greater than . Our construction disproves a conjecture of Scott that
graphs excluding induced subdivisions of any fixed graph have chromatic number
bounded by a function of their clique number.Comment: Small corrections, bibliography updat
Triangle-free geometric intersection graphs with no large independent sets
It is proved that there are triangle-free intersection graphs of line
segments in the plane with arbitrarily small ratio between the maximum size of
an independent set and the total number of vertices.Comment: Change of the title, minor revisio
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 curves that cross a fixed curve
We prove that for every integer , the class of intersection graphs
of curves in the plane each of which crosses a fixed curve in at least one and
at most points is -bounded. This is essentially the strongest
-boundedness result one can get for this kind of graph classes. As a
corollary, we prove that for any fixed integers and , every
-quasi-planar topological graph on vertices with any two edges crossing
at most times has edges.Comment: Small corrections, improved presentatio
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
Decomposition of multiple packings with subquadratic union complexity
Suppose is a positive integer and is a -fold packing of
the plane by infinitely many arc-connected compact sets, which means that every
point of the plane belongs to at most sets. Suppose there is a function
with the property that any members of determine
at most holes, which means that the complement of their union has at
most bounded connected components. We use tools from extremal graph
theory and the topological Helly theorem to prove that can be
decomposed into at most (-fold) packings, where is a constant
depending only on and .Comment: Small generalization of the main result, improvements in the proofs,
minor correction
Triangle-free geometric intersection graphs with large chromatic number
Several classical constructions illustrate the fact that the chromatic number
of a graph can be arbitrarily large compared to its clique number. However,
until very recently, no such construction was known for intersection graphs of
geometric objects in the plane. We provide a general construction that for any
arc-connected compact set in that is not an axis-aligned
rectangle and for any positive integer produces a family of
sets, each obtained by an independent horizontal and vertical scaling and
translation of , such that no three sets in pairwise intersect
and . This provides a negative answer to a question of
Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to
construct a triangle-free family of homothetic (uniformly scaled) copies of a
set with arbitrarily large chromatic number. This applies to many common
shapes, like circles, square boundaries, and equilateral L-shapes.
Additionally, we reveal a surprising connection between coloring geometric
objects in the plane and on-line coloring of intervals on the line.Comment: Small corrections, bibliography updat
Coloring triangle-free rectangle overlap graphs with colors
Recently, it was proved that triangle-free intersection graphs of line
segments in the plane can have chromatic number as large as . Essentially the same construction produces -chromatic
triangle-free intersection graphs of a variety of other geometric
shapes---those belonging to any class of compact arc-connected sets in
closed under horizontal scaling, vertical scaling, and
translation, except for axis-parallel rectangles. We show that this
construction is asymptotically optimal for intersection graphs of boundaries of
axis-parallel rectangles, which can be alternatively described as overlap
graphs of axis-parallel rectangles. That is, we prove that triangle-free
rectangle overlap graphs have chromatic number , improving on
the previous bound of . To this end, we exploit a relationship
between off-line coloring of rectangle overlap graphs and on-line coloring of
interval overlap graphs. Our coloring method decomposes the graph into a
bounded number of subgraphs with a tree-like structure that "encodes"
strategies of the adversary in the on-line coloring problem. Then, these
subgraphs are colored with colors using a combination of
techniques from on-line algorithms (first-fit) and data structure design
(heavy-light decomposition).Comment: Minor revisio
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