A successful concept for measuring non-planarity of graphs: The crossing number

Discrete Mathematics

Volumn 276, Issue 1-3, 2004, Pages 331-352

  Székely, László A ^a  

^a UNIVERSITY OF SOUTH CAROLINA   (United States)

Author keywords

Crossing number; Graph drawing

Indexed keywords

GRAPH THEORY; NUMBER THEORY; PROBLEM SOLVING; THEOREM PROVING;

CROSSING NUMBER; GRAPH DRAWING;

COMPUTATIONAL GEOMETRY;

EID: 0346154872     PISSN: 0012365X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0012-365X(03)00317-0     Document Type: Conference Paper

References (73)

1. Aigner M., Ziegler G.M. Proofs from the Book. 1998;Springer, Berlin.
2. Ajtai M., Chvátal V., Newborn M., Szemerédi E. Crossing-free subgraphs. Ann. Discrete Math. 12:1982;9-12.
3. 6 , in: Proceedings of the 27th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, 1996; Congr. Numer. 118 (1996) 97-107.
4. 6 , in: Proceedings of the 27th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Baton Rouge, LA, 1996; Congr. Numer. 118 (1996) 97-107.
5. 7 , in: Proceedings of the 28th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, 1997, Congr. Numer. 125 (1997) 97-117.
6. 7 , in: Proceedings of the 28th Southeastern International Conference on Combinatorics, Graph Theory and Computing, Boca Raton, FL, 1997, Congr. Numer. 125 (1997) 97-117.
7. Beineke L.W., Ringeisen R.D. On the crossing number of product of cycles and graphs of order four. J. Graph Theory. 4:1980;145-155.
8. Bienstock D., Dean N. New results on rectilinear crossing numbers and plane embeddings. J. Graph Theory. 16:1992;389-398.
9. Bienstock D., Dean N. Bounds on the rectilinear crossing numbers. J. Graph Theory. 17:1993;333-348.
10. J. Blažek, N. Koman, A minimal problem concerning complete plane graphs, in: M. Fiedler (Ed.), Theory of Graphs and its Applications, Proceedings of the Symposium on Smolenice, 1963, Publ. House of the Czechoslovak Academy of Sciences, Prague, 1964, pp. 113-117.
11. Busacker R.G., Saaty T.L. Finite Graphs and Networks: an Introduction with Applications. 1967;McGraw-Hill, New York, Toronto, London.
12. Chojnacki C. Über wesentliche unplättbare Kurven in dreidimensionalen Raume. Fund. Math. 23:1934;135-142.
13. Dey T. Improved bounds for planar. k -sets and related problems Discrete Comput. Geom. 19:1998;373-382.
14. Eggleton R.B. Rectilinear drawings of graphs. Utilitas Math. 29:1986;149-172.
15. Elekes G. On the number of sums and products. Acta Arithm. 81(4):1997;365-367.
16. Elekes G., Nathanson M.B., Ruzsa I.Z. Convexity and sumsets. J. Number Theory. 83(2):2000;194-201.
17. Elekes G., Rónyai L. A combinatorial problem on polynomials and rational functions. J. Combin. Theory A. 89:2000;1-20.
18. Erdos P., Guy R.P. Crossing number problems. Amer. Math. Monthly. 80:1973;52-58.
19. G. Even, S. Guha, B. Schieber, Improved approximations of crossings in graph drawings and VLSI layout areas, in: Proceedings of the 32nd Annual Symposium on Theory of Computing, STOC'00, ACM Press, New York, 2000, pp. 296-305.
20. Exoo G., Harary F., Kabell J. The crossing numbers of some generalized Petersen graphs. Math. Scand. 48:1981;184-188.
21. Fáry I. On straight line representations of graphs. Acta Univ. Szeged Sect. Sci. Math. 11:1948;229-233.
22. S. Fiorini, On the crossing number of generalized Petersen graphs, in: Combinatorics '84, Bari, 1984, North-Holland Mathematical Studies, Vol. 123, North-Holland, Amsterdam, 1986, pp. 225-241.
23. Garey M.R., Johnson D.S. Crossing number is NP-complete. SIAM J. Algebra Discrete Methods. 4:1983;312-316.
24. n) = (m-2)n is true for all but finitely n , for each m , to appear.
25. Granville A., Roesler F. The set of differences of a given set. Amer. Math. Monthly. 106(4):1999;338-344.
26. Guy R.K. The decline and fall of Zarankiewicz's theorem. Harary F. Proof Techniques in Graph Theory. 1969;63-69 Academic Press, New York, London.
27. Guy R.K. Crossing number of graphs. Alavi Y., et al. Graph Theory and Applications. Lecture Notes in Mathematics. Vol. 303:1972;111-124 Springer, New York.
28. Guy R.K. Math. Rev. 58:1974;21749.
29. Harary F., Kainen P.C., Schwenk A.J. Toroidal graphs with arbitrarily high crossing numbers. Nanta Math. 6:1973;58-67.
30. n Časopis Pěst. Mat. 107:1982;225-230.
31. Jensen F. An upper bound for the rectilinear crossing number of complete graphs. J. Combin. Theory. 11:1971;212-216.
32. 5,n J. Combin. Theory. 9:1970;315-323.
33. Klešč M. The crossing numbers of products of paths and stars with 4-vertex graphs. J. Graph Theory. 18:1994;605-614.
34. Klešč M. The crossing numbers of products of paths with 5-vertex graphs. Discrete Math. 223:2001;353-359.
35. n J. Graph Theory. 22:1996;239-243.
36. 1 -norm of an exponential sum, in: Teoriya priblizheniy funkciy i operatorov, Tezisy dokladov Mezhdunarodnoi konferencii, posvyashchennoi 80-letiyu so dnya rozhdeniya Sergeia Borisovicha Stechkina, Ekaterinburg, 2000, pp. 88-89 (in Russian).
37. Lakatos I. Worrall J., Zahar E. Proofs and Reputations: the Logic of Mathematical Discovery. 1976;Cambridge University Press, Cambridge, New York, Melbourne.
38. Leighton F.T. Complexity Issues in VLSI. 1983;MIT Press, Cambridge.
39. F.T. Leighton, S. Rao, An approximate max flow min cut theorem for multicommodity flow problem with applications to approximation algorithm, in: Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Washington, DC, 1988, pp. 422-431; J. ACM 46 (1999) 787-832.
40. F.T. Leighton, S. Rao, An approximate max flow min cut theorem for multicommodity flow problem with applications to approximation algorithm, in: Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Washington, DC, 1988, pp. 422-431; J. ACM 46 (1999) 787-832.
41. V.F. Lev, MR 99j:11022.
42. LovrečičSaražin M. The crossing number of the generalized Petersen graph. P(10,4) is four Math. Slovaca. 47:1997;189-192.
43. Mansfield A. Determining the thickness of graphs is NP-hard. Math. Proc. Cambridge Philos. Soc. 9:1983;9-23.
44. McQuillan D., Richter R.B. On the crossing numbers of certain generalized Petersen graphs. Discrete Math. 104:1992;311-320.
45. Mohar B. Problem Mentioned at the Special Session on Topological Graph Theory. 1995;Mathfest, Burlington, Vermont.
46. Owens A. On the biplanar crossing number. IEEE Trans. Circuit Theory. CT-18:1971;277-280.
47. Pach J. Crossing numbers. Akiyama J., Kano M., Urabe M. Discrete and Computational Geometry Japanese Conference JCDCG'98, Tokyo, Japan, December 1998. Lecture Notes in Computer Science. Vol. 1763:2000;267-273 Springer, Berlin.
48. Pach J., Sharir M. On the number of incidences between points and curves. Combin. Probab. Comput. 7:1998;121-127.
49. J. Pach, J. Spencer, G. Tóth, New bounds on crossing numbers, 15th ACM Symposium on Computational Geometry, ACM, 1999, pp. 124-133; Discrete Comput. Geom. 24 (2000) 623-644.
50. J. Pach, J. Spencer, G. Tóth, New bounds on crossing numbers, 15th ACM Symposium on Computational Geometry, ACM, 1999, pp. 124-133; Discrete Comput. Geom. 24 (2000) 623-644.
51. Pach J., Tóth G. Graphs drawn with few crossings per edge. Combinatorica. 17:1997;427-439.
52. J. Pach, G. Tóth, Which crossing number is it anyway? in: Proceedings of the 39th Annual Symposium on Foundation of Computer Science, IEEE Press, Baltimore, 1998, pp. 617-626; J. Combin. Theory Ser. B 80 (2000) 225-246.
53. J. Pach, G. Tóth, Which crossing number is it anyway? in: Proceedings of the 39th Annual Symposium on Foundation of Computer Science, IEEE Press, Baltimore, 1998, pp. 617-626; J. Combin. Theory Ser. B 80 (2000) 225-246.
54. Pach J., Tóth G. Thirteen problems on crossing numbers. Geombinatorics. 9:2000;194-207.
55. 5 Discrete Comput. Geom. 13:1995;149-159.
56. Richter R.B., Thomassen C. Relations between crossing numbers of complete and complete bipartite graphs. Amer. Math. Monthly. 104:1997;131-137.
57. n J. Combin. Theory. 24:1978;134-134.
58. N. Robertson, P.D. Seymour, Excluding a graph with one crossing, in: Graph Structure Theory, Seattle, WA, 1991, Contemporary Mathematics, Vol. 147, American Mathematical Society, Providence, RI, 1993, pp. 669-675.
59. Shahrokhi F., Sýkora O., Székely L.A., Vrťo I. The crossing number of a graph on a compact 2-manifold. Adv. Math. 123:1996;105-119.
60. F. Shahrokhi, O. Sýkora, L.A. Székely, I. Vrťo, Drawings of graphs on surfaces with few crossings, Algorithmica (16) 1996 118-131.
61. Shahrokhi F., Sýkora O., Székely L.A., Vrťo I. Crossing numbers: bounds and applications. Bárány I., Böröczky K. Intuitive Geometry. Bolyai Society Mathematical Studies. Vol. 6:1997;179-206 János Bolyai Mathematical Society, Budapest.
62. J. Spencer, G. Tóth, Crossing number of random graphs, to appear.
63. O. Sýkora, L.A. Székely, I. Vrťo, Two counterexamples in graph drawing, in: Proc. 27th Intl. Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, Vol. 2573, Springer, Berlin, 2002.
64. O. Sýkora, L.A. Székely, I. Vrťo, Crossing numbers and biplanar crossing numbers I: a survey, in preparation.
65. O. Sýkora, L.A. Székely, I. Vrťo, Crossing numbers and biplanar crossing numbers II: using the probabilistic method, in preparation.
66. Székely L.A. Crossing numbers and hard Erdos problems in discrete geometry. Combin. Probab. Comput. 6:1997;353-358.
67. Turán P. A note of welcome. J. Graph Theory. 1:1977;7-9.
68. Tutte W.T. Toward a theory of crossing numbers. J. Combin. Theory. 8:1970;45-53.
69. Urbaník K. Solution du problème posé par P. Turán. Colloq. Math. 3:1955;200-201.
70. I. Vrťo, Crossing Numbers of Graphs: A Bibliography, http:/sun.ifi.savba.sk/̃imrich/.
71. A.T. White, L.W. Beineke, Topological graph theory, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Topics in Graph Theory, Academic Press, 1978, pp. 15-50.
72. Woodall D.R. Cyclic-order graphs and Zarankiewicz's crossing-number conjecture. J. Graph Theory. 17:1993;657-671.
73. Zarankiewicz K. On a problem of P. Turán concerning graphs. Fund. Math. 41:1954;137-145.