research-article

Coloring k_k-free intersection graphs of geometric objects in the plane

Authors:

János PachAuthors Info & Claims

SCG '08: Proceedings of the twenty-fourth annual symposium on Computational geometry

Pages 346 - 354

https://doi.org/10.1145/1377676.1377735

Published: 09 June 2008 Publication History

Abstract

The intersection graph of a collection C of sets is a graph on the vertex set C, in which C₁,C₂ ∈ C are joined by an edge if and only if C₁ ∩ C₂ ≠ Ø. Erdös 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 Kk-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 n0 such that every topological graph with n ≥ n0 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.

References

[1]

E. Ackerman, On the maximum number of edges in topological graphswith no four pairwise crossing edges, Proc. 22nd ACM Symp. onComputational Geometry (SoCG), Sedona, AZ, June 2006, 259--263.]]

Digital Library

[2]

P.K. Agarwal, B. Aronov, J. Pach, R. Pollack, and M. Sharir,Quasi-planar graphs have a linear number of edges, Proc. Graph Drawing '95, Passau (Franz J. Brandenburg, ed.), Lecture Notes in Computer Science 1027, Springer-Verlag, Berlin, 1996,1--7.]]

Digital Library

[3]

.K. Agarwal, M. van Kreveld, and S. Suri, Label placement bymaximum independent set in rectangles, Comput. Geom. Theory Appl. 11 (1998), 209--218.]]

Digital Library

[4]

P. K. Agarwal and N. H. Mustafa, Independent set of intersectiongraphs of convex objects in 2D, Comput. Geometry: Theory Appls. 34 (2006), 83--95.]]

Digital Library

[5]

R. Ahlswede and I. Karapetyan, Intersection graphs of rectangles and segments, in: General Theory of Information Transfer and Combinatorics, Lecture Notes in Comp. Science 4123, Springer-Verlag, Berlin, 2006, 1064--1065.]]

Digital Library

[6]

B. Aronov, P. Erdös, W. Goddard, D. Kleitman, M. Klugerman, J.Pach, and L. J. Schulman, Crossing families. Combinatorica14 (1994), 127--134.]]

[7]

E. Asplund and B. Grünbaum, On a coloring problem, Math. Scand., 8 (1960), 181--188.]]

[8]

S. Basu, R. Pollack, and M.-F. Roy, Algorithms in realalgebraic geometry. Second edition. Algorithms and Computation in Mathematics 10, Springer-Verlag, Berlin, 2006.]]

Digital Library

[9]

P. Berman, B. DasGupta, S. Muthukrishnan, and S. Ramaswami, Improved approximation algorithms for rectangle tiling and packing, in: Proc. 12th ACM-SIAM Sympos. Discrete Algorithms, 2001, 427--436.]]

Digital Library

[10]

T. M. Chan, Polynomial-time approximation schemes for packing and piercing fat objects, J. Algorithms 46 (2003), 178--189.]]

Digital Library

[11]

J. Fox, A bipartite analogue of Dilworth's theorem, Order 23 (2006), 197--209.]]

[12]

J. Fox and J. Pach, Separator theorems and Turán-type results forplanar intersection graphs, Advances in Mathematics, submitted.]]

[13]

J. Fox and J. Pach, String graphs and incomparability graphs, inpreparation.]]

[14]

J. Fox and J. Pach, Erdös-Hajnal-type results on intersectionpatterns of geometric objects, Bolyai Soc. Mathematical Studies, J. Bolyai Mathematical Soc., Budapest, to appear.]]

[15]

J. Fox, J. Pach, and Cs. Tóth, Intersection patterns of curves, J. London Mathematical Society, accepted.]]

[16]

J. Fox, J. Pach, and Cs. Tóth, Turán-type results for partialorders and intersection graphs of convex sets, Israel J.Math., submitted.]]

[17]

R. J. Fowler, M. S. Paterson, and S.L. Tanimoto, Optimal packing and covering in the plane are NP-complete, Inform. Process. Lett. 12 (1981), 133--137.]]

[18]

M. R. Garey and D. S. Johnson, Computers andIntractability: A Guide to the Theory of NP--Completeness, Freeman,New York, 1979.]]

Digital Library

[19]

M. Goljan, J. Kratochvíl, and P. Kučera, String graphs, Rozpravy Ceskoslovenské Akad. Ved Rada Mat. Prírod. Ved 96 (1986), 96 pp.]]

[20]

M. C. Golumbic, D. Rotem, and J. Urrutia, Comparability graphs and intersection graphs. Discrete Math. 43 (1983), 37--46.]]

Digital Library

[21]

A. Gyárfás, Problems from the world surrounding perfect graphs. Zastos. Mat. 19 (1987), 413--441 (1988).]]

[22]

A. Gyárfás and J. Lehel, Covering and coloring problems forrelatives of intervals, Discrete Math., 55 (1985),167--180.]]

[23]

J. Hastad, Clique is hard to approximate within n<sup>1--ε</sup>, Acta Math. 182 (1999), 105--142.]]

[24]

D. S. Hochbaum and W. Maass, Approximation schemes for covering and packing problems in image processing and VLSI, J. ACM 32(1985), 130--136.]]

Digital Library

[25]

H. Imai and T. Asano, Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane, J. Algorithms 4 (1983), 310--323.]]

[26]

S. Khanna, S. Muthukrishnan and M. Paterson, On approximating rectangle tiling and packing, in: Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, 1998,384--393]]

Digital Library

[27]

S.-R. Kim, A. V. Kostochka, and K. Nakprasit, On the chromaticnumber of intersection graphs of convex sets in the plane, Electron. J. Combin. 11 (2004), Research Paper 52, 12 pp.]]

[28]

P. Koebe, Kontaktprobleme der konformen Abbildung, Berichteüber die Verhandlungen der Sachsischen Akademie derWissenschaften, Leipzig, Mathematische-Physische Klasse 88,(1936), 141--164.]]

[29]

P. Kolman and J. Matoušek, Crossing number, pair-crossing number,and expansion. J. Combin. Theory Ser. B 92 (2004), 99--113.]]

Digital Library

[30]

A. V. Kostochka, Coloring intersection graphs of geometric figureswith a given clique number, Towards a theory of geometric graphs, Contemp. Math., 342, Amer. Math. Soc., Providence, RI,(2004), 127--138.]]

[31]

A. V. Kostochka and J. Nešetril, Coloring relatives of intervals on the plane. I. Chromatic number versus girth. European J. Combin. 19 (1998), 103--110.]]

Digital Library

[32]

A. V. Kostochka and J. Nešetril, Colouring relatives of intervals on the plane. II. Intervals and rays in two directions. European J. Combin. 23 (2002),37--41.]]

Digital Library

[33]

D. Larman, J. Matoušek, J. Pach, and J. Töröcsik, ARamsey-type result for convex sets, Bull. London Math. Soc.26 (1994), 132--136.]]

[34]

R. J. Lipton and R. E. Tarjan, A separator theorem for planargraphs. Proceedings of a Conference on Theoretical Computer Science (Univ. Waterloo, Waterloo, Ont., 1977), pp. 1--10. Comput.Sci. Dept., Univ. Waterloo, Waterloo, Ont., 1978.]]

[35]

F. Maffray and M. Preissmann, On the NP-completeness of the-colorability problem for triangle-free graphs, Discrete Mathematics 162 (1996), 313--317.]]

Digital Library

[36]

S. McGuinness, Colouring arcwise connected sets in the plane. I. Graphs Combin. 16 (2000), 429--439.]]

[37]

G. L. Miller, S.-H. Teng, W. Thurston, and S. A. Vavasis, Separators for sphere-packings and nearest neighbor graphs, J. ACM 44 (1) (1997), 1--29.]]

Digital Library

[38]

J. Pach, Notes on geometric graph theory, Discrete and Computational Geometry (J.E. Goodman et al, eds.), DIMACS Series, Vol 6, Amer. Math. Soc., Providence, 1991, 273--285.]]

[39]

J. Pach, R. Radoičić, and G. Tóth, Relaxing planarity fortopological graphs, Discrete and Computational Geometry (J.Akiyama, M. Kano, eds.), Lecture Notes in Computer Science 2866,Springer-Verlag, Berlin, 2003, 221--232.]]

[40]

J. Pach, F. Shahrokhi, and M. Szegedy, Applications of the crossing number, Proc. 10th ACM Symposium on Computational Geometry, 1994, 198--202.]]

Digital Library

[41]

J. Pach and J. Töröcsik, Some geometric applicationsof Dilworth's theorem, Discrete Comput. Geom. 12 (1994),1--7.]]

Digital Library

[42]

J. Pach and G. Tóth, Unavoidable configurations in complete topological graphs, Graph Drawing 2000, Lecture Notes in Computer Science 1984, Springer-Verlag, 2001, 328--337.]]

Digital Library

[43]

J. Pach and G. Tóth, Comment on Fox News, Geombinatorics 15 (2006), 150--154.]]

[44]

R. N. Shmatkov, On coloring polygon-circle graphs with clique number2, Siberian Adv. Math. 10 (2000), 73--86.]]

[45]

J. B. Sidney, S. J. Sidney, and J. Urrutia, Circle orders, n-gon orders and the crossing number, Order 5(1) (1988), 1--10.]]

[46]

W. D. Smith and N. C. Wormald, Geometric separator theorems and applications, in: 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1998, 232--243.]]

Digital Library

[47]

P. Turán, On a problem in extremal graph theory (in Hungarian). Math. Fiz. Lapok 48 (1941), 436--452.]]

[48]

P. Valtr, On geometric graphs with no pairwise parallel edges. Dedicated to the memory of Paul Erdös. Discrete Comput.Geom. 19 (1998), no. 3, Special Issue, 461--469.]]

[49]

P. Valtr, Graphs drawn in the plane with no pairwise crossing edges, In G. D. Battista, editor, Graph Drawing, volume 1353 of Lecture Notes in Computer Science, 205--218, Springer, 1997.]]

Digital Library

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Angelini PDa Lozzo GFörster HSchneck T(2021)2-Layer k-Planar GraphsGraph Drawing and Network Visualization10.1007/978-3-030-68766-3_32(403-419)Online publication date: 14-Feb-2021
https://doi.org/10.1007/978-3-030-68766-3_32
Suk A(2013)Disjoint Edges in Complete Topological GraphsDiscrete & Computational Geometry10.1007/s00454-012-9481-x49:2(280-286)Online publication date: 1-Mar-2013
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Fox JPach JDey TWhitesides S(2012)String graphs and incomparability graphsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261311(405-414)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261311
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Index Terms

Coloring k_k-free intersection graphs of geometric objects in the plane
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
      1. Graph algorithms

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cover image ACM Conferences

SCG '08: Proceedings of the twenty-fourth annual symposium on Computational geometry

June 2008

304 pages

ISBN:9781605580715

DOI:10.1145/1377676

Program Chair:
Monique Teillaud
INRIA Sophia Antipolis - Méditerranée, France

Copyright © 2008 ACM.

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Published: 09 June 2008

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Cited By

Angelini PDa Lozzo GFörster HSchneck T(2021)2-Layer k-Planar GraphsGraph Drawing and Network Visualization10.1007/978-3-030-68766-3_32(403-419)Online publication date: 14-Feb-2021
https://doi.org/10.1007/978-3-030-68766-3_32
Suk A(2013)Disjoint Edges in Complete Topological GraphsDiscrete & Computational Geometry10.1007/s00454-012-9481-x49:2(280-286)Online publication date: 1-Mar-2013
https://dl.acm.org/doi/10.1007/s00454-012-9481-x
Fox JPach JDey TWhitesides S(2012)String graphs and incomparability graphsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261311(405-414)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261311
Suk ADey TWhitesides S(2012)Disjoint edges in complete topological graphsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261308(383-386)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261308
Di Giacomo EDidimo WLiotta GMontecchiani F(2012)h-quasi planar drawings of bounded treewidth graphs in linear areaProceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science10.1007/978-3-642-34611-8_12(91-102)Online publication date: 26-Jun-2012
https://dl.acm.org/doi/10.1007/978-3-642-34611-8_12
Cardinal JKorman M(2012)Coloring planar homothets and three-dimensional hypergraphsProceedings of the 10th Latin American international conference on Theoretical Informatics10.1007/978-3-642-29344-3_11(121-132)Online publication date: 16-Apr-2012
https://dl.acm.org/doi/10.1007/978-3-642-29344-3_11
Suk A(2011)k-Quasi-planar graphsProceedings of the 19th international conference on Graph Drawing10.1007/978-3-642-25878-7_26(266-277)Online publication date: 21-Sep-2011
https://dl.acm.org/doi/10.1007/978-3-642-25878-7_26
Flier HMihal$#225;k MWidmayer PZych A(2011)Maximum independent set in 2-direction outersegment graphsProceedings of the 37th international conference on Graph-Theoretic Concepts in Computer Science10.1007/978-3-642-25870-1_15(155-166)Online publication date: 21-Jun-2011
https://dl.acm.org/doi/10.1007/978-3-642-25870-1_15
Pach JSuk ATreml MKirkpatrick DMitchell J(2010)Tangencies between families of disjoint regions in the planeProceedings of the twenty-sixth annual symposium on Computational geometry10.1145/1810959.1811029(423-428)Online publication date: 13-Jun-2010
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Ackerman EFox JPach JSuk AHershberger JFogel E(2009)On grids in topological graphsProceedings of the twenty-fifth annual symposium on Computational geometry10.1145/1542362.1542430(403-412)Online publication date: 8-Jun-2009
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