Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Abstract

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 R 2 that is not an axis-aligned rectangle and for any positive integer k produces a family F of sets, each obtained by an independent horizontal and vertical scaling and translation of X , such that no three sets in F pairwise intersect and χ(F) > k. This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

Coloring Curves that Cross a Fixed Curve

“…A significant part of this research has been devoted to understanding classes of string graphs that are χ-bounded, which means that every graph G in the class satisfies χ(G) f (ω(G)) for some function f : N → N. Here, χ (G) and ω(G) denote the chromatic number and the clique number (the maximum size of a clique) of G, respectively. Recently, Pawlik et al [24,25] proved that the class of all string graphs is not χ-bounded. However, all known constructions of string graphs with small clique number and large chromatic number require a lot of freedom in placing curves around in the plane.…”

“…The class of string graphs is not χ-bounded. Pawlik et al [24,25] showed that Burling's construction for boxes in R 3 can be adapted to provide a construction of triangle-free intersection graphs of straight-line segments (or geometric shapes of various other kinds) with chromatic number growing as fast as Θ(log log n) with the number of vertices n. It was further generalized to a construction of string graphs with clique number ω and chromatic number Θ ω ((log log n) ω−1 ) [16]. The best known upper bound on the chromatic number of string graphs in terms of the number of vertices is (log n) O(log ω) , proved by Fox and Pach [8] using a separator theorem for string graphs due to Matoušek [18].…”

“…The proof of Theorem 6 is an easy adaptation of the construction from [24,25] and is presented in detail in Section 4. The rest of this section is devoted to the proof of Theorem 4.…”

Restricted Frame Graphs and a Conjecture of Scott

“…replacing every edge of G by a path on at least k + 1 edges. A recent result [10] (see [9] for a follow-up) shows that Scott's conjecture is false whenever H is a ≥ 1-subdivision of a non-planar graph. The proof is based on a construction of a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (the existence of such graphs also disproved a conjecture of Erdős).…”

“…Recently, Walczak [13] showed how to slightly modify the construction of [10,9] to obtain a family of graphs with no stable sets of linear size (in particular, with unbounded fractional chromatic number). Therefore, the following stronger result can be deduced: for any non-planar graph H, there exist graphs with no ≥1-subdivision of H as an induced subgraph, and with no stable sets of linear size.…”