15,764 research outputs found

    Triangle-free geometric intersection graphs with no large independent sets

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    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.Comment: Change of the title, minor revisio

    Triangle-free geometric intersection graphs with large chromatic number

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    Several classical constructions illustrate the fact that the chromatic number of a graph can 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 XX in R2\mathbb{R}^2 that is not an axis-aligned rectangle and for any positive integer kk produces a family F\mathcal{F} of sets, each obtained by an independent horizontal and vertical scaling and translation of XX, such that no three sets in F\mathcal{F} pairwise intersect and χ(F)>k\chi(\mathcal{F})>k. This provides a negative answer to a question of Gyarfas 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, and 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.Comment: Small corrections, bibliography updat

    Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

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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 XX X in R2\mathbb{R }^2 R 2 that is not an axis-aligned rectangle and for any positive integer kk k produces a family F\mathcal{F } F of sets, each obtained by an independent horizontal and vertical scaling and translation of XX X , such that no three sets in F\mathcal{F } F pairwise intersect and χ(F)>k\chi (\mathcal{F })>k 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 lin

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

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    In the 1970s, Erdos 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 kk, we construct a triangle-free family of line segments in the plane with chromatic number greater than kk. 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.Comment: Small corrections, bibliography updat

    Decomposition of multiple packings with subquadratic union complexity

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    Suppose kk is a positive integer and X\mathcal{X} is a kk-fold packing of the plane by infinitely many arc-connected compact sets, which means that every point of the plane belongs to at most kk sets. Suppose there is a function f(n)=o(n2)f(n)=o(n^2) with the property that any nn members of X\mathcal{X} determine at most f(n)f(n) holes, which means that the complement of their union has at most f(n)f(n) bounded connected components. We use tools from extremal graph theory and the topological Helly theorem to prove that X\mathcal{X} can be decomposed into at most pp (11-fold) packings, where pp is a constant depending only on kk and ff.Comment: Small generalization of the main result, improvements in the proofs, minor correction

    Coloring triangle-free rectangle overlap graphs with O(loglogn)O(\log\log n) colors

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    Recently, it was proved that triangle-free intersection graphs of nn line segments in the plane can have chromatic number as large as Θ(loglogn)\Theta(\log\log n). Essentially the same construction produces Θ(loglogn)\Theta(\log\log n)-chromatic triangle-free intersection graphs of a variety of other geometric shapes---those belonging to any class of compact arc-connected sets in R2\mathbb{R}^2 closed under horizontal scaling, vertical scaling, and translation, except for axis-parallel rectangles. We show that this construction is asymptotically optimal for intersection graphs of boundaries of axis-parallel rectangles, which can be alternatively described as overlap graphs of axis-parallel rectangles. That is, we prove that triangle-free rectangle overlap graphs have chromatic number O(loglogn)O(\log\log n), improving on the previous bound of O(logn)O(\log n). To this end, we exploit a relationship between off-line coloring of rectangle overlap graphs and on-line coloring of interval overlap graphs. Our coloring method decomposes the graph into a bounded number of subgraphs with a tree-like structure that "encodes" strategies of the adversary in the on-line coloring problem. Then, these subgraphs are colored with O(loglogn)O(\log\log n) colors using a combination of techniques from on-line algorithms (first-fit) and data structure design (heavy-light decomposition).Comment: Minor revisio

    Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

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    We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient (1+ε)(1+\varepsilon)-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density

    Applications of a new separator theorem for string graphs

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    An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the present note, this bound is combined with a result of the authors, according to which every dense string graph contains a large complete balanced bipartite graph. Three applications are given concerning string graphs G with n vertices: (i) if K_t is not a subgraph of G for some t, then the chromatic number of G is at most (\log n)^{O(\log t)}; (ii) if K_{t,t} is not a subgraph of G, then G has at most t(\log t)^{O(1)}n edges,; and (iii) a lopsided Ramsey-type result, which shows that the Erdos-Hajnal conjecture almost holds for string graphs.Comment: 7 page

    Coloring curves that cross a fixed curve

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    We prove that for every integer t1t\geq 1, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most tt points is χ\chi-bounded. This is essentially the strongest χ\chi-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers k2k\geq 2 and t1t\geq 1, every kk-quasi-planar topological graph on nn vertices with any two edges crossing at most tt times has O(nlogn)O(n\log n) edges.Comment: Small corrections, improved presentatio
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