Disjointness graphs of short polygonal chains
J\'anos Pach, G\'abor Tardos, G\'eza T\'oth

TL;DR
This paper investigates the chromatic properties of disjointness graphs of polygonal chains, showing limitations for length 2 chains and establishing bounds for length 3 chains, revealing complex coloring behaviors.
Contribution
It demonstrates that disjointness graphs of length 2 polygonal chains are not always χ-bounded and provides bounds for length 3 chains, advancing understanding of geometric graph coloring.
Findings
Disjointness graphs of length 2 chains are not χ-bounded.
Constructed length 3 chains with large girth and chromatic number.
Established χ-boundedness for length 3 chains in infinite cases.
Abstract
The {\em disjointness graph} of a set system is a graph whose vertices are the sets, two being connected by an edge if and only if they are disjoint. It is known that the disjointness graph of any system of segments in the plane is {\em -bounded}, that is, its chromatic number is upper bounded by a function of its clique number . Here we show that this statement does not remain true for systems of polygonal chains of length . We also construct systems of polygonal chains of length such that their disjointness graphs have arbitrarily large girth and chromatic number. In the opposite direction, we show that the class of disjointness graphs of (possibly self-intersecting) \emph{-way infinite} polygonal chains of length is -bounded: for every such graph , we have
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Advanced Graph Theory Research
