946 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 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
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
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
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