Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

Walczak, Bartosz

doi:10.1007/s00454-014-9645-y

Cited by 6 publications

(7 citation statements)

References 8 publications

(12 reference statements)

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“…These subdivisions do not appear as induced subgraphs in the construction, so it follows that such graphs H are counterexamples to Scott's conjecture. It turns out that graphs in the modified construction of Walczak [13] can be obtained from graphs in Pawlik et al's construction [10,9,6] by adding twins. As the class of restricted frame graphs is stable by the operation of twin addition (see Remark 2.5), the graphs H we find are also counterexamples to a weaker version of Scott's conjecture, where the chromatic number is replaced by the fractional chromatic number.…”

Section: Our Resultsmentioning

confidence: 99%

“…While it might seem restrictive to study graphs that cannot be represented as restricted frame graphs (instead of graphs that do not appear in the construction), it can be proven that in the case of ≥2-subdivisions of multigraphs, this is not restrictive at all: for every multigraph G, if some ≥2-subdivision H of G can be represented as a restricted frame graph, then H appears as an induced subgraph in the construction of Pawlik et al [10,9,6] and in the modified construction of Walczak [13]. So the construction can be thought of as universal for ≥2-subdivisions of restricted frame graphs.…”

Section: Our Resultsmentioning

confidence: 99%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Restricted Frame Graphs and a Conjecture of Scott

Chalopin

Mendez

2016

Electron. J. Combin.

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Scott proved in 1997 that for any tree T , every graph with bounded clique number which does not contain any subdivision of T as an induced subgraph has bounded chromatic number. Scott also conjectured that the same should hold if T is replaced by any graph H. Pawlik et al. recently constructed a family of triangle-free intersection graphs of segments in the plane with unbounded chromatic number (thereby disproving an old conjecture of Erdős). This shows that Scott's conjecture is false whenever H is obtained from a non-planar graph by subdividing every edge at least once.It remains interesting to decide which graphs H satisfy Scott's conjecture and which do not. In this paper, we study the construction of Pawlik et al. in more details to extract more counterexamples to Scott's conjecture. For example, we show that Scott's conjecture is false for any graph obtained from K 4 by subdividing every edge at least once. We also prove that if G is a 2-connected multigraph with no vertex contained in every cycle of G, then any graph obtained from G by subdividing every edge at least twice is a counterexample to Scott's conjecture. *

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Section: Our Resultsmentioning

confidence: 99%

Section: Our Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Restricted Frame Graphs and a Conjecture of Scott

Chalopin

Mendez

2016

Electron. J. Combin.

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“…If G is a family of intersection or disjointness graphs of certain nice shapes (e.g. disks, boxes, segments, convex sets, curves), then either the chromatic number of every member can be bounded by a function of the clique number [4,9,13,17,19,23,25,26,28] (in which case G is called a χ-bounded family), or the chromatic number grows as at most a polylogarithmic function of the number of vertices assuming the clique number is fixed [8,16,22,27,28,34,35,38,39]. In particular, a general result of Fox and Pach [16] shows that if G is the intersection graph of n arc-wise connected sets in the plane (also known as a string graph), then χ(G) ≤ (log n) O(log ω(G)) .…”

Section: Coloring Linesmentioning

confidence: 99%

Geometric constructions: lines and boxes

Tomon¹

2023

Preprint

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The goal of this paper is to show the existence (using probabilistic tools) of configurations of lines, boxes, and points with certain interesting combinatorial properties.(i) First, we construct a family of n lines in R 3 whose intersection graph is triangle-free of chromatic number Ω(n 1/15 ). This improves the previously best known bound Ω(log log n) by Norin, and is also the first construction of a triangle-free intersection graph of simple geometric objects with polynomial chromatic number.(ii) Furthermore, we construct a set of n points P and a family of n axis-parallel boxes B in R d such that the incidence graph of (P, B) is K 2,2 -free of average degree at least (log n) d−1−o(1) . This also provides an n-vertex graph of separation dimension 2d and average degree (log n) d−1−o(1) . The former answers a question of Basit, Chernikov, Starchenko, Tao, and Tran, while the latter addresses a problem of Alon, Basavaraju, Chandran, Mathew, and Rajendraprasad.(iii) Finally, we construct a set of n points in R d , whose Delaunay graph with respect to axis-parallel boxes has independence number at most n • (log n) −(d−1)/2+o(1) . This extends the planar case considered by Chen, Pach, Szegedy, and Tardos.

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“…In 2014, Pawlik, Kozik, Krawczyk, Lasoń, Micek, Trotter, and B. Walczak [PaKK14] represented a class of K 3 -free graphs originally constructed by Burling [Bu65] as segment graphs whose chromatic numbers can be arbitrarily large. Shortly after, Walczak [Wa15] strengthened this result by proving that there are K 3 -free segment graphs on n vertices in which every independent set is of size O( n log log n ). Using the same approach, in order to prove Conjecture 1.1 for some r, it would be sufficient to show that there is a constant g(r) with the property that the vertex set of every K r -free string graph can be colored by g(r) colors such that each (string) graph induced by one of the color classes is K 4 -free.…”

Section: Introductionmentioning

confidence: 99%

Quasiplanar graphs, string graphs, and the Erdos-Gallai problem

Fox¹,

Pach²,

Suk³

2021

Preprint

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An r-quasiplanar graph is a graph drawn in the plane with no r pairwise crossing edges. We prove that there is a constant C > 0 such that for any s > 2, every 2 s -quasiplanar graph with n vertices has at most n( C log n s ) 2s−4 edges. A graph whose vertices are continuous curves in the plane, two being connected by an edge if and only if they intersect, is called a string graph. We show that for every ǫ > 0, there exists δ > 0 such that every string graph with n vertices, whose chromatic number is at least n ǫ contains a clique of size at least n δ . A clique of this size or a coloring using fewer than n ǫ colors can be found by a polynomial time algorithm in terms of the size of the geometric representation of the set of strings.In the process, we use, generalize, and strengthen previous results of Lee, Tomon, and others. All of our theorems are related to geometric variants of the following classical graph-theoretic problem of Erdős, Gallai, and Rogers. Given a K r -free graph on n vertices and an integer s < r, at least how many vertices can we find such that the subgraph induced by them is K s -free?

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Triangle-Free Geometric Intersection Graphs with No Large Independent Sets

Abstract: It is proved that there are triangle-free intersection graphs of line segments in the plane with arbitrarily small ratio between the maximum size of an independent set and the total number of vertices.

Cited by 6 publications

References 8 publications

Restricted Frame Graphs and a Conjecture of Scott

Restricted Frame Graphs and a Conjecture of Scott

Geometric constructions: lines and boxes

Quasiplanar graphs, string graphs, and the Erdos-Gallai problem

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