Coloring graphs with geometric representation
Jarosław Grytczuk Jarek Grytczuk
EuroGIGA GraDR midterm meeting, Berlin, 2012
Segments in the plane
Question (1970): Is it true that the chromatic number of
triangle-free intersection graphs of segments in the plane
is bounded?
Segments in the plane
Theorem (Kozik, Krawczyk, Lason, Micek, Pawlik, Trotter, Walczak):
The chromatic number of a triangle-free graph of
segments in the plane can be arbitrarily large.
Proof: Recursive construction resembling (but much
more sophisticated) that of Mycielski.
Problem: What is the maximum chromatic number of
an n-vertex triangle-free graph of segments in the plane?
Ω(log log n) – lower bound
Ο(log n) - upper bound (McGuinness, 2000)
Forbidden subdivisions
Conjecture (Scott, 1999): Let H be a fixed graph. Then
triangle-free graphs not containing any subdivision of H as
an induced subgraph have bounded chromatic number.
Observation (Fox, Pach):
The negative solution for segments in the plane
disproves Scott's conjecture when H is a 1-subdivision
of any non-planar graph.
Axis-aligned rectangles
Problem (Bielecki, 1948): Is it
true that triangle-free graphs
of axis-aligned rectangles have
bounded chromatic number?
Theorem (Asplund, Grünbaum, 1960):
Yes, it is at most six (and six is best possible). In general,
the chromatic number of rectangles is at most 4k2 – 3k,
when the clique number of a related graph is at most k.
Problem: Is the above theorem true with linear number of
colors?
Axes-aligned frames
Theorem (Kozik, Krawczyk, Lason, Micek, Pawlik, Trotter, Walczak):
The chromatic number of
a triangle-free graph of
axis-aligned frames
can be arbitrarily large.
Theorem (Krawczyk, Pawlik, Walczak):
The maximum chromatic number of an n-vertex
triangle-free graph of frames is asymptotically equal to
log log n.
Deformations by independent scaling
(x,y) → (ax+b,cy+d)
Theorem (Kozik, Krawczyk, Lason, Micek, Pawlik, Trotter, Walczak):
Let S be fixed subset of the plane which is compact and
arcwise connected. Assume also that S is not an axisaligned rectangle. Then the chromatic number of a
triangle-free intersection graph of deformations of S may
be arbitrarily large.
L-shapes
Question (Gyarfas, Lehel): Is it
true that the chromatic number
of triangle-free intersection
graphs of L-shapes in the plane
is bounded?
No, by the previous theorem.
Disks and Circles
Theorem (Kim, Kostochka, Nakprasit):
Let F be a fixed compact covex set in the plane.
Then the chromatic number of a family of
homothetic copies of F with clique number k is at
most 6k – 6.
Theorem (Kozik, Krawczyk, Lason, Micek, Pawlik, Trotter, Walczak):
The chromatic number of
a triangle-free graph of
circles in the plane
can be arbitrarily large.
Ringel's circle problem
Problem: Is it true that „triangle-free” contact graphs
of circles have bounded chromatic number?
Thank You!