11,684 research outputs found
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 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
Hard and Easy Instances of L-Tromino Tilings
We study tilings of regions in the square lattice with L-shaped trominoes.
Deciding the existence of a tiling with L-trominoes for an arbitrary region in
general is NP-complete, nonetheless, we identify restrictions to the problem
where it either remains NP-complete or has a polynomial time algorithm. First,
we characterize the possibility of when an Aztec rectangle and an Aztec diamond
has an L-tromino tiling. Then, we study tilings of arbitrary regions where only
rotations of L-trominoes are available. For this particular case we
show that deciding the existence of a tiling remains NP-complete; yet, if a
region does not contains certain so-called "forbidden polyominoes" as
sub-regions, then there exists a polynomial time algorithm for deciding a
tiling.Comment: Full extended version of LNCS 11355:82-95 (WALCOM 2019
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