1,273 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
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
Outerstring graphs are -bounded
An outerstring graph is an intersection graph of curves that lie in a common
half-plane and have one endpoint on the boundary of that half-plane. We prove
that the class of outerstring graphs is -bounded, which means that their
chromatic number is bounded by a function of their clique number. This
generalizes a series of previous results on -boundedness of outerstring
graphs with various additional restrictions on the shape of curves or the
number of times the pairs of curves can cross. The assumption that each curve
has an endpoint on the boundary of the half-plane is justified by the known
fact that triangle-free intersection graphs of straight-line segments can have
arbitrarily large chromatic number.Comment: Introduction extended by a survey of results on (outer)string graphs,
some 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
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
Restricted frame graphs and a conjecture of Scott
Scott proved in 1997 that for any tree , every graph with bounded clique
number which does not contain any subdivision of as an induced subgraph has
bounded chromatic number. Scott also conjectured that the same should hold if
is replaced by any graph . Pawlik et al. recently constructed a family
of triangle-free intersection graphs of segments in the plane with unbounded
chromatic number (thereby disproving an old conjecture of Erd\H{o}s). This
shows that Scott's conjecture is false whenever is obtained from a
non-planar graph by subdividing every edge at least once.
It remains interesting to decide which graphs satisfy Scott's conjecture
and which do not. In this paper, we study the construction of Pawlik et al. in
more details to extract more counterexamples to Scott's conjecture. For
example, we show that Scott's conjecture is false for any graph obtained from
by subdividing every edge at least once. We also prove that if is a
2-connected multigraph with no vertex contained in every cycle of , then any
graph obtained from by subdividing every edge at least twice is a
counterexample to Scott's conjecture.Comment: 21 pages, 8 figures - Revised version (note that we moved some of our
results to an appendix
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
Burling graphs, chromatic number, and orthogonal tree-decompositions
A classic result of Asplund and Gr\"unbaum states that intersection graphs of
axis-aligned rectangles in the plane are -bounded. This theorem can be
equivalently stated in terms of path-decompositions as follows: There exists a
function such that every graph that has two
path-decompositions such that each bag of the first decomposition intersects
each bag of the second in at most vertices has chromatic number at most
. Recently, Dujmovi\'c, Joret, Morin, Norin, and Wood asked whether this
remains true more generally for two tree-decompositions. In this note we
provide a negative answer: There are graphs with arbitrarily large chromatic
number for which one can find two tree-decompositions such that each bag of the
first decomposition intersects each bag of the second in at most two vertices.
Furthermore, this remains true even if one of the two decompositions is
restricted to be a path-decomposition. This is shown using a construction of
triangle-free graphs with unbounded chromatic number due to Burling, which we
believe should be more widely known.Comment: v3: minor changes made following comments by the referees, v2: minor
edit
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
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