Triangle-free intersection graphs of line segments with large chromatic number

Abstract

In the 1970s, Erdős 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 k, we construct a triangle-free family of line segments in the plane with chromatic number greater than k. 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. arXiv:1209.1595v5 [math.CO]

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

“…Recently [12], we showed that the answer is negative. Namely, for every positive integer k we construct a family F of line segments in the plane with no three pairwise intersecting segments and such that χ(F) > k.…”

“…We already posed several problems concerning the chromatic number and independence number of triangle-free segment intersection graphs in [12]. Here we focus on classification of shapes with respect to whether triangle-free families of copies of the shape under particular transformations have bounded or unbounded chromatic number.…”

Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

“…It is also shown in [7] how the graph G k is represented as a segment intersection graph. I will show that there is an assignment w k of positive integer weights to the vertices of G k with the following properties:…”

“…Pawlik et al [7] define also a graphG k , which arises from (G k , P k ) by adding, for every probe P ∈ P k , a diagonal d P connected to all vertices in P. This is the smallest triangle-free segment intersection graph known to have chromatic number greater than k. Define the assignmentw k of weights to the vertices ofG k so thatw k is equal to w k on the vertices of G k andw k (d P ) = 1 for every P ∈ P k . Let I be an independent set inG k .…”

The Beginnings of Geometric Graph Theory

“…The corresponding edges form a planar subgraph of G, therefore we would obtain |E(G)|/c k ≤ 3n − 6, where n ≥ 3 denotes the number of vertices of G. This would imply |E(G)| = O k (n), as required. However, Pawlik, Kozik, Krawczyk, Lasoń, Miczek, Trotter, and Walczak [PawK12] constructed systems of n segments, no 3 of which are pairwise intersecting, such that they cannot be decomposed into fewer than log log n subsystems of disjoint segments. Therefore, the answer to Erdős' question is no.…”