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Coloring Triangle-Free Rectangle Overlap Graphs with \(O(\log \log n)\) Colors

Tomasz Krawczyk Arkadiusz Pawlik Bartosz Walczak Recently, it was proved that triangle-free intersection graphs of n line segments in the plane can have chromatic number as large as ( log log n

Coloring Intersection Graphs of Arc-Connected Sets in the Plane

A family of sets in the plane is simple if the intersection of any subfamily is arc-connected, and it is pierced by a line \(L\) if the intersection of any member with \(L\) is a nonempty segment. It is proved that the intersection graphs of simple families of compact arc-connected sets in the plane pierced by a common line have chromatic number bounded by a function of their ...

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may 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 \(X\) in \(\mathbb{R }^2\) that is not an ...