On the chromatic number of disjointness graphs of curves

Pach, János; Tomon, István

doi:10.1016/j.jctb.2020.02.003

Cited by 15 publications

(17 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the rest of the section, we prove part (2) of Theorem 4. For the proof, we use the following characterization of intersection graphs of x-monotone curves that intersect the same vertical line, which was established in [15].…”

Section: Sharp Threshold For Intersection Graphs-proof Of Theoremmentioning

confidence: 99%

Ordered graphs and large bi-cliques in intersection graphs of curves

Pach

Tomon

2019

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

An ordered graph G < is a graph with a total ordering < on its vertex set. A monotone path of length k is a sequence of vertices v 1 < v 2 < . . . < v k such that v i v j is an edge of G < if and only if |j − i| = 1. A bi-clique of size m is a complete bipartite graph whose vertex classes are of size m.We prove that for every positive integer k, there exists a constant c k > 0 such that every ordered graph on n vertices that does not contain a monotone path of length k as an induced subgraph has a vertex of degree at least c k n, or its complement has a bi-clique of size at least c k n/ log n. A similar result holds for ordered graphs containing no induced ordered subgraph isomorphic to a fixed ordered matching.As a consequence, we give a short combinatorial proof of the following theorem of Fox and Pach. There exists a constant c > 0 such the intersection graph G of any collection of n x-monotone curves in the plane has a bi-clique of size at least cn/ log n or its complement contains a bi-clique of size at least cn. (A curve is called x-monotone if every vertical line intersects it in at most one point.) We also prove that if G has at most 1 4 − ǫ n 2 edges for some ǫ > 0, then G contains a linear sized bi-clique. We show that this statement does not remain true if we replace 1 4 by any larger constants.

show abstract

Section: Sharp Threshold For Intersection Graphs-proof Of Theoremmentioning

confidence: 99%

Ordered graphs and large bi-cliques in intersection graphs of curves

Pach

Tomon

2019

European Journal of Combinatorics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Something similar can be done, however, for general x-monotone curves. Inspired by the present paper, Pach and Tomon [14] very recently constructed a set of curves whose disjointness graph has large chromatic number, and therefore cannot be covered with fewer than four comparability graphs. It would be interesting to decide if the four comparability graphs are needed for segments, as well.…”

Section: Discussionmentioning

confidence: 96%

Improved Ramsey-type results for comparability graphs

Korándi

Tomon

2020

Combinator. Probab. Comp.

Self Cite

View full text Add to dashboard Cite

Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$ . This bound is known to be tight for $r=1$ . The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for $r=2$ , as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$ . Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size $cn/{(log n)^r}$ . With this, we improve a result of Fox and Pach.

show abstract

“…The same is true for systems x-monotone curves, that is, for continuous curves in the plane with the property that every vertical line intersects them in at most one point. It was shown in [PT20] that, in this generality, the order of magnitude of this bound cannot be improved. On the other hand, we proved [PTT17] that the class of disjointness graphs of strings (continuous curves in the plane) is not χ-bounded.…”

Section: Introductionmentioning

confidence: 98%

Disjointness graphs of short polygonal chains

Pach¹,

Tardos²,

Tóth³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The 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 G of any system of segments in the plane is χ-bounded, that is, its chromatic number χ(G) is upper bounded by a function of its clique number ω(G).Here we show that this statement does not remain true for systems of polygonal chains of length 2. We also construct systems of polygonal chains of length 3 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) 2-way infinite polygonal chains of length 3 is χ-bounded: for every such graph G, we have χ(G) ≤ (ω(G)) 3 +ω(G).

show abstract

On the chromatic number of disjointness graphs of curves

Cited by 15 publications

References 40 publications

Ordered graphs and large bi-cliques in intersection graphs of curves

Ordered graphs and large bi-cliques in intersection graphs of curves

Improved Ramsey-type results for comparability graphs

Disjointness graphs of short polygonal chains

Contact Info

Product

Resources

About