Crossing number (graph theory) (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Crossing number (graph theory)" in English language version.

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25doi.org

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1tilburguniversity.edu

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1combinatorics.org

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mathscinet.ams.org

Guy, Richard K. (1969). "The decline and fall of Zarankiewicz's theorem". Proof Techniques in Graph Theory (Proc. Second Ann Arbor Graph Theory Conf., Ann Arbor, Mich., 1968). Academic Press, New York. pp. 63–69. MR 0253931..
Pan, Shengjun; Richter, R. Bruce (2007). "The crossing number of $K 11$ is $100$ ". Journal of Graph Theory. 56 (2): 128–134. doi:10.1002/jgt.20249. MR 2350621. S2CID 37155969..
Clancy, Kieran; Haythorpe, Michael; Newcombe, Alex; Pegg, Ed Jr. (2020). "There are no cubic graphs on 26 vertices with crossing number 10 or 11". Graphs and Combinatorics. 36 (6): 1713–1721. arXiv:1804.10336. doi:10.1007/s00373-020-02204-6. MR 4163409. S2CID 253889498.
Garey, M. R.; Johnson, D. S. (1983). "Crossing number is NP-complete". SIAM Journal on Algebraic and Discrete Methods. 4 (3): 312–316. doi:10.1137/0604033. MR 0711340.
Hliněný, P. (2006). "Crossing number is hard for cubic graphs". Journal of Combinatorial Theory. Series B. 96 (4): 455–471. doi:10.1016/j.jctb.2005.09.009. MR 2232384.
Grohe, M. (2005). "Computing crossing numbers in quadratic time". Journal of Computer and System Sciences. 68 (2): 285–302. arXiv:cs/0009010. doi:10.1016/j.jcss.2003.07.008. MR 2059096.
Ajtai, M.; Chvátal, V.; Newborn, M.; Szemerédi, E. (1982). Crossing-free subgraphs. Theory and Practice of Combinatorics. North-Holland Mathematics Studies. Vol. 60. pp. 9–12. MR 0806962.
Székely, L. A. (1997). "Crossing numbers and hard Erdős problems in discrete geometry". Combinatorics, Probability and Computing. 6 (3): 353–358. doi:10.1017/S0963548397002976. MR 1464571. S2CID 36602807.
Dey, T. K. (1998). "Improved bounds for planar k-sets and related problems". Discrete and Computational Geometry. 19 (3): 373–382. doi:10.1007/PL00009354. MR 1608878.
Ringel, Gerhard (1965). "Ein Sechsfarbenproblem auf der Kugel". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German). 29 (1–2): 107–117. doi:10.1007/BF02996313. MR 0187232. S2CID 123286264.

archive.today

Oswin Aichholzer. "Rectilinear Crossing Number project". Archived from the original on 2012-12-30., and Rectilinear Crossing Number on the Institute for Software Technology at Graz, University of Technology (2009).

arxiv.org

de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, B.; Salazar, G. (2006). "Improved bounds for the crossing numbers of K_m,n and K_n". SIAM Journal on Discrete Mathematics. 20 (1): 189–202. arXiv:math/0404142. doi:10.1137/S0895480104442741. S2CID 1509054.
Barát, János; Tóth, Géza (2009). "Towards the Albertson Conjecture". arXiv:0909.0413 [math.CO].
Clancy, Kieran; Haythorpe, Michael; Newcombe, Alex; Pegg, Ed Jr. (2020). "There are no cubic graphs on 26 vertices with crossing number 10 or 11". Graphs and Combinatorics. 36 (6): 1713–1721. arXiv:1804.10336. doi:10.1007/s00373-020-02204-6. MR 4163409. S2CID 253889498.
Cabello S. and Mohar B. (2013). "Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard". SIAM Journal on Computing. 42 (5): 1803–1829. arXiv:1203.5944. doi:10.1137/120872310. S2CID 6535755.
Grohe, M. (2005). "Computing crossing numbers in quadratic time". Journal of Computer and System Sciences. 68 (2): 285–302. arXiv:cs/0009010. doi:10.1016/j.jcss.2003.07.008. MR 2059096.

combinatorics.org

Schaefer, Marcus (2014). "The graph crossing number and its variants: a survey". The Electronic Journal of Combinatorics. 1000. DS21. doi:10.37236/2713.

depaul.edu

ovid.cs.depaul.edu

Schaefer, Marcus (2010). Complexity of some geometric and topological problems (PDF). Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers. Lecture Notes in Computer Science. Vol. 5849. Springer-Verlag. pp. 334–344. doi:10.1007/978-3-642-11805-0_32. ISBN 978-3-642-11804-3..

doi.org

Purchase, Helen C.; Cohen, Robert F.; James, Murray I. (1995). "Validating graph drawing aesthetics". In Brandenburg, Franz-Josef (ed.). Graph Drawing, Symposium on Graph Drawing, GD '95, Passau, Germany, September 20-22, 1995, Proceedings. Lecture Notes in Computer Science. Vol. 1027. Springer. pp. 435–446. doi:10.1007/BFb0021827. ISBN 978-3-540-60723-6..
Turán, P. (1977). "A Note of Welcome". Journal of Graph Theory. 1: 7–9. doi:10.1002/jgt.3190010105.
Bronfenbrenner, Urie (1944). "The graphic presentation of sociometric data". Sociometry. 7 (3): 283–289. doi:10.2307/2785096. JSTOR 2785096. The arrangement of subjects on the diagram, while haphazard in part, is determined largely by trial and error with the aim of minimizing the number of intersecting lines.
Scheinerman, Edward R.; Wilf, Herbert S. (1994). "The rectilinear crossing number of a complete graph and Sylvester's "four point problem" of geometric probability". American Mathematical Monthly. 101 (10): 939–943. doi:10.2307/2975158. JSTOR 2975158.
Pach, J.; Tóth, G. (1998). "Which crossing number is it, anyway?". Proceedings of the 39th Annual Symposium on Foundations of Computer Science (FOCS 1998). pp. 617–626. doi:10.1109/SFCS.1998.743512..
de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, B.; Salazar, G. (2006). "Improved bounds for the crossing numbers of K_m,n and K_n". SIAM Journal on Discrete Mathematics. 20 (1): 189–202. arXiv:math/0404142. doi:10.1137/S0895480104442741. S2CID 1509054.
Zarankiewicz, K. (1954). "On a Problem of P. Turán Concerning Graphs". Fundamenta Mathematicae. 41: 137–145. doi:10.4064/fm-41-1-137-145.
Saaty, T.L. (1964). "The minimum number of intersections in complete graphs". Proceedings of the National Academy of Sciences of the United States of America. 52 (3): 688–690. Bibcode:1964PNAS...52..688S. doi:10.1073/pnas.52.3.688. PMC 300329. PMID 16591215.
Pan, Shengjun; Richter, R. Bruce (2007). "The crossing number of $K 11$ is $100$ ". Journal of Graph Theory. 56 (2): 128–134. doi:10.1002/jgt.20249. MR 2350621. S2CID 37155969..
Pegg, E. T.; Exoo, G. (2009). "Crossing Number Graphs". Mathematica Journal. 11 (2). doi:10.3888/tmj.11.2-2.
Clancy, Kieran; Haythorpe, Michael; Newcombe, Alex; Pegg, Ed Jr. (2020). "There are no cubic graphs on 26 vertices with crossing number 10 or 11". Graphs and Combinatorics. 36 (6): 1713–1721. arXiv:1804.10336. doi:10.1007/s00373-020-02204-6. MR 4163409. S2CID 253889498.
Bhatt, Sandeep N.; Leighton, Frank Thomson (1984-04-01). "A framework for solving VLSI graph layout problems". Journal of Computer and System Sciences. 28 (2): 300–343. doi:10.1016/0022-0000(84)90071-0. ISSN 0022-0000.
Garey, M. R.; Johnson, D. S. (1983). "Crossing number is NP-complete". SIAM Journal on Algebraic and Discrete Methods. 4 (3): 312–316. doi:10.1137/0604033. MR 0711340.
Hliněný, P. (2006). "Crossing number is hard for cubic graphs". Journal of Combinatorial Theory. Series B. 96 (4): 455–471. doi:10.1016/j.jctb.2005.09.009. MR 2232384.
Cabello S. and Mohar B. (2013). "Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard". SIAM Journal on Computing. 42 (5): 1803–1829. arXiv:1203.5944. doi:10.1137/120872310. S2CID 6535755.
Schaefer, Marcus (2010). Complexity of some geometric and topological problems (PDF). Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers. Lecture Notes in Computer Science. Vol. 5849. Springer-Verlag. pp. 334–344. doi:10.1007/978-3-642-11805-0_32. ISBN 978-3-642-11804-3..
Grohe, M. (2005). "Computing crossing numbers in quadratic time". Journal of Computer and System Sciences. 68 (2): 285–302. arXiv:cs/0009010. doi:10.1016/j.jcss.2003.07.008. MR 2059096.
Kawarabayashi, Ken-ichi; Reed, Bruce (2007). Computing crossing number in linear time. Proceedings of the 29th Annual ACM Symposium on Theory of Computing. pp. 382–390. doi:10.1145/1250790.1250848. ISBN 978-1-59593-631-8.
Even, Guy; Guha, Sudipto; Schieber, Baruch (2003). "Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas". SIAM Journal on Computing. 32 (1): 231–252. doi:10.1137/S0097539700373520.
Székely, L. A. (1997). "Crossing numbers and hard Erdős problems in discrete geometry". Combinatorics, Probability and Computing. 6 (3): 353–358. doi:10.1017/S0963548397002976. MR 1464571. S2CID 36602807.
Dey, T. K. (1998). "Improved bounds for planar k-sets and related problems". Discrete and Computational Geometry. 19 (3): 373–382. doi:10.1007/PL00009354. MR 1608878.
Bienstock, Daniel; Dean, Nathaniel (July 1993). "Bounds for rectilinear crossing numbers". Journal of Graph Theory. 17 (3): 333–348. doi:10.1002/jgt.3190170308.
Ringel, Gerhard (1965). "Ein Sechsfarbenproblem auf der Kugel". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German). 29 (1–2): 107–117. doi:10.1007/BF02996313. MR 0187232. S2CID 123286264.
Schaefer, Marcus (2014). "The graph crossing number and its variants: a survey". The Electronic Journal of Combinatorics. 1000. DS21. doi:10.37236/2713.

dx.doi.org

Bhatt, Sandeep N.; Leighton, Frank Thomson (1984-04-01). "A framework for solving VLSI graph layout problems". Journal of Computer and System Sciences. 28 (2): 300–343. doi:10.1016/0022-0000(84)90071-0. ISSN 0022-0000.

haifa.ac.il

sci.haifa.ac.il

Ackerman, Eyal (2013). "On topological graphs with at most four crossings per edge" (PDF). Archived from the original (PDF) on 2014-07-14.

harvard.edu

ui.adsabs.harvard.edu

Saaty, T.L. (1964). "The minimum number of intersections in complete graphs". Proceedings of the National Academy of Sciences of the United States of America. 52 (3): 688–690. Bibcode:1964PNAS...52..688S. doi:10.1073/pnas.52.3.688. PMC 300329. PMID 16591215.

indstate.edu

isu.indstate.edu

Exoo, G. "Rectilinear Drawings of Famous Graphs".

jstor.org

Bronfenbrenner, Urie (1944). "The graphic presentation of sociometric data". Sociometry. 7 (3): 283–289. doi:10.2307/2785096. JSTOR 2785096. The arrangement of subjects on the diagram, while haphazard in part, is determined largely by trial and error with the aim of minimizing the number of intersecting lines.
Scheinerman, Edward R.; Wilf, Herbert S. (1994). "The rectilinear crossing number of a complete graph and Sylvester's "four point problem" of geometric probability". American Mathematical Monthly. 101 (10): 939–943. doi:10.2307/2975158. JSTOR 2975158.

nih.gov

ncbi.nlm.nih.gov

Saaty, T.L. (1964). "The minimum number of intersections in complete graphs". Proceedings of the National Academy of Sciences of the United States of America. 52 (3): 688–690. Bibcode:1964PNAS...52..688S. doi:10.1073/pnas.52.3.688. PMC 300329. PMID 16591215.

pubmed.ncbi.nlm.nih.gov

Saaty, T.L. (1964). "The minimum number of intersections in complete graphs". Proceedings of the National Academy of Sciences of the United States of America. 52 (3): 688–690. Bibcode:1964PNAS...52..688S. doi:10.1073/pnas.52.3.688. PMC 300329. PMID 16591215.

semanticscholar.org

api.semanticscholar.org

de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, B.; Salazar, G. (2006). "Improved bounds for the crossing numbers of K_m,n and K_n". SIAM Journal on Discrete Mathematics. 20 (1): 189–202. arXiv:math/0404142. doi:10.1137/S0895480104442741. S2CID 1509054.
Pan, Shengjun; Richter, R. Bruce (2007). "The crossing number of $K 11$ is $100$ ". Journal of Graph Theory. 56 (2): 128–134. doi:10.1002/jgt.20249. MR 2350621. S2CID 37155969..
Clancy, Kieran; Haythorpe, Michael; Newcombe, Alex; Pegg, Ed Jr. (2020). "There are no cubic graphs on 26 vertices with crossing number 10 or 11". Graphs and Combinatorics. 36 (6): 1713–1721. arXiv:1804.10336. doi:10.1007/s00373-020-02204-6. MR 4163409. S2CID 253889498.
Cabello S. and Mohar B. (2013). "Adding One Edge to Planar Graphs Makes Crossing Number and 1-Planarity Hard". SIAM Journal on Computing. 42 (5): 1803–1829. arXiv:1203.5944. doi:10.1137/120872310. S2CID 6535755.
Székely, L. A. (1997). "Crossing numbers and hard Erdős problems in discrete geometry". Combinatorics, Probability and Computing. 6 (3): 353–358. doi:10.1017/S0963548397002976. MR 1464571. S2CID 36602807.
Ringel, Gerhard (1965). "Ein Sechsfarbenproblem auf der Kugel". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German). 29 (1–2): 107–117. doi:10.1007/BF02996313. MR 0187232. S2CID 123286264.

tilburguniversity.edu

research.tilburguniversity.edu

de Klerk, E.; Maharry, J.; Pasechnik, D. V.; Richter, B.; Salazar, G. (2006). "Improved bounds for the crossing numbers of K_m,n and K_n". SIAM Journal on Discrete Mathematics. 20 (1): 189–202. arXiv:math/0404142. doi:10.1137/S0895480104442741. S2CID 1509054.

tugraz.at

ist.tugraz.at

Oswin Aichholzer. "Rectilinear Crossing Number project". Archived from the original on 2012-12-30., and Rectilinear Crossing Number on the Institute for Software Technology at Graz, University of Technology (2009).

dist.ist.tugraz.at

Oswin Aichholzer. "Rectilinear Crossing Number project". Archived from the original on 2012-12-30., and Rectilinear Crossing Number on the Institute for Software Technology at Graz, University of Technology (2009).

web.archive.org

Ackerman, Eyal (2013). "On topological graphs with at most four crossings per edge" (PDF). Archived from the original (PDF) on 2014-07-14.

wiley.com

onlinelibrary.wiley.com

Bienstock, Daniel; Dean, Nathaniel (July 1993). "Bounds for rectilinear crossing numbers". Journal of Graph Theory. 17 (3): 333–348. doi:10.1002/jgt.3190170308.

wolfram.com

mathworld.wolfram.com

Weisstein, Eric W. "Graph Crossing Number". MathWorld.

worldcat.org

search.worldcat.org

Bhatt, Sandeep N.; Leighton, Frank Thomson (1984-04-01). "A framework for solving VLSI graph layout problems". Journal of Computer and System Sciences. 28 (2): 300–343. doi:10.1016/0022-0000(84)90071-0. ISSN 0022-0000.