Covering and coloring problems for relatives of intervals

Gyárfás, András; Lehel, Jenö

doi:10.1016/0012-365x(85)90045-7

Cited by 77 publications

(75 citation statements)

References 7 publications

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“…Our result also gives a negative answer to the question of Gyárfás and Lehel [7] whether the families of axis-aligned L-shapes, that is, shapes consisting of a horizontal and a vertical segments of arbitrary lengths forming the letter 'L', are χ -bounded. We prove that this is not the case even for equilateral L-shapes.…”

Section: It Is Clear That χ(G) ω(G)mentioning

confidence: 74%

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Pawlik

Kozik

Krawczyk

et al. 2013

Discrete Comput Geom

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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.

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Section: It Is Clear That χ(G) ω(G)mentioning

confidence: 74%

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Pawlik

Kozik

Krawczyk

et al. 2013

Discrete Comput Geom

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“…Since then, this topic has received considerable attention [5,22,27,30,31,32,36,44]. In particular, a classical question of Erdős [21,31,36] asks whether the chromatic number of all triangle-free intersection graphs of segments in the plane is bounded by a constant.…”

Section: Introductionmentioning

confidence: 99%

Coloring k _k -free intersection graphs of geometric objects in the plane

Fox

Pach

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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The intersection graph of a collection C of sets is a graph on the vertex set C, in which C 1 , C 2 ∈ C are joined by an edge if and only if C 1 ∩ C 2 = ∅. Erdős conjectured that the chromatic number of triangle-free intersection graphs of n segments in the plane is bounded from above by a constant. Here we show that it is bounded by a polylogarithmic function of n, which is the first nontrivial bound for this problem. More generally, we prove that for any t and k, the chromatic number of every K k -free intersection graph of n curves in the plane, every pair of which have at most t points in common, is at most (c t log n log k ) c log k , where c is an absolute constant and c t only depends on t. We establish analogous results for intersection graphs of convex sets, x-monotone curves, semialgebraic sets of constant description complexity, and sets that can be obtained as the union of a bounded number of sets homeomorphic to a disk.Using a mix of results on partially ordered sets and planar separators, for large k we improve the best known upper bound on the number of edges of a k-quasi-planar topological graph with n vertices, that is, a graph drawn in the plane with curvilinear edges, no k of which are pairwise crossing. As another application, we show that for every ε > 0 and for every positive integer t, there exist δ > 0 and a positive integer n 0 such that every topological graph with n ≥ n 0 vertices, at least n 1+ε edges, and no pair of edges intersecting in more than t points, has at least n δ pairwise intersecting edges.

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“…A graph is called an overlap graph if its vertices can be represented by intervals on a line such that two vertices are adjacent if and only if the corresponding intervals overlap but neither contains the other [1]. Gy~rf~s [2] (see also [3]) has shown that every triangle-free overlap graph can be colored by a constant number, c, of colors, and Kostochka [4] proved that this is true with c = 5.…”

Section: (E) Incident To a Graph Vertex In V Is Also O(iei)mentioning

confidence: 99%