34 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
Piercing axis-parallel boxes
Let \F be a finite family of axis-parallel boxes in such that \F
contains no pairwise disjoint boxes. We prove that if \F contains a
subfamily \M of pairwise disjoint boxes with the property that for every
F\in \F and M\in \M with , either contains a
corner of or contains corners of , then \F can be
pierced by points. One consequence of this result is that if and
the ratio between any of the side lengths of any box is bounded by a constant,
then \F can be pierced by points. We further show that if for each two
intersecting boxes in \F a corner of one is contained in the other, then \F
can be pierced by at most points, and in the special case
where \F contains only cubes this bound improves to
Coloring translates and homothets of a convex body
We obtain improved upper bounds and new lower bounds on the chromatic number
as a linear function of the clique number, for the intersection graphs (and
their complements) of finite families of translates and homothets of a convex
body in \RR^n.Comment: 11 pages, 2 figure
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 the complements of intersection graphs of geometric figures
AbstractLet G¯ be the complement of the intersection graph G of a family of translations of a compact convex figure in Rn. When n=2, we show that χ(G¯)⩽min{3α(G)-2,6γ(G)}, where γ(G) is the size of the minimum dominating set of G. The bound on χ(G¯)⩽6γ(G) is sharp. For higher dimension we show that χ(G¯)⩽⌈2(n2-n+1)1/2⌉n-1⌈(n2-n+1)1/2⌉(α(G)-1)+1, for n⩾3. We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in Rn