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| [[File:3-crossing Heawood graph.svg|thumb|A drawing of the [[Heawood graph]] with three crossings. This is the minimum number of crossings among all drawings of this graph, so the graph has crossing number cr(''G'') = 3.]]
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| In [[graph theory]], the '''crossing number''' cr(''G'') of a graph ''G'' is the lowest number of edge crossings of a plane [[graph drawing|drawing]] of the graph ''G''. For instance, a graph is [[planar graph|planar]] [[if and only if]] its crossing number is zero.
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| The mathematical origin of the study of crossing numbers is in
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| '''Turán's brick factory problem''', in which [[Pál Turán]] asked to determine the crossing number of the [[complete bipartite graph]] ''K''<sub>''m,n''</sub>.<ref>{{cite journal |doi=10.1002/jgt.3190010105 |last=Turán |first=P. |title=A Note of Welcome |journal=J. Graph Theory |volume=1 |pages=7–9 |year=1977 |ref=harv}}</ref> However, the same problem of minimizing crossings was also considered in [[sociology]] at approximately the same time as Turán, in connection with the construction of [[sociogram]]s.<ref>{{cite journal|title=The graphic presentation of sociometric data|first=Urie|last=Bronfenbrenner|authorlink=Urie Bronfenbrenner|journal=Sociometry|volume=7|issue=3|year=1944|pages=283–289|jstor=2785096|quote=The arrangement of subjects on
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| the diagram, while haphazard in part, is determined largely by trial and error with the aim of minimizing the number of intersecting lines.}}</ref> It continues to be of great importance in [[graph drawing]].
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| Without further qualification, the crossing number allows drawings in which the edges may be represented by arbitrary curves; the '''rectilinear crossing number''' requires all edges to be straight line segments, and may differ from the crossing number. In particular, the rectilinear crossing number of a [[complete graph]] is essentially the same as the minimum number of convex quadrilaterals determined by a set of ''n'' points in general position, closely related to the [[Happy Ending problem]].<ref>{{Cite journal |title=The rectilinear crossing number of a complete graph and Sylvester's "four point problem" of geometric probability |first1=Edward R. |last1=Scheinerman|author1-link=Ed Scheinerman |first2=Herbert S. |last2=Wilf|author2-link=Herbert Wilf |journal=[[American Mathematical Monthly]] |volume=101 |issue=10 |year=1994 |pages=939–943 |doi=10.2307/2975158 |jstor=2975158 |ref=harv}}</ref>
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| ==History==
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| During [[World War II]], Hungarian mathematician [[Pál Turán]] was forced to work in a brick factory, pushing wagon loads of bricks from kilns to storage sites. The factory had tracks from each kiln to each storage site, and the wagons were harder to push at the points where tracks crossed each other, from which Turán was led to ask his brick factory problem: what is the minimum possible number of crossings in a drawing of a [[complete bipartite graph]]?<ref>{{cite book
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| | last1 = Pach | first1 = János | author1-link = János Pach
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| | last2 = Sharir | first2 = Micha | author2-link = Micha Sharir
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| | contribution = 5.1 Crossings—the Brick Factory Problem
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| | pages = 126–127
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| | publisher = [[American Mathematical Society]]
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| | series = Mathematical Surveys and Monographs
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| | title = Combinatorial Geometry and Its Algorithmic Applications: The Alcalá Lectures
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| | volume = 152
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| | year = 2009}}</ref>
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| [[Kazimierz Zarankiewicz|Zarankiewicz]] attempted to solve Turán's brick factory problem;<ref>{{cite journal |last=Zarankiewicz |first=K. |title=On a Problem of P. Turán Concerning Graphs |journal=Fund. Math. |volume=41 |pages=137–145 |year=1954 |ref=harv}}</ref> his proof contained an error,
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| but he established a valid upper bound of
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| :<math>cr(K_{m,n}) \le \left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right\rfloor\left\lfloor\frac{m}{2}\right\rfloor\left\lfloor\frac{m-1}{2}\right\rfloor</math>
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| for the crossing number of the complete bipartite graph ''K<sub>m,n</sub>''. The conjecture that this inequality is actually an equality is now known as Zarankiewicz' crossing number conjecture. The gap in the proof of the lower bound was not discovered until eleven years after publication, nearly simultaneously by [[Gerhard Ringel]] and [[Paul Chester Kainen|Paul Kainen]]; see <ref name="ptig">{{Cite journal |title=Decline and fall of Zarankiewicz's Theorem |first1=R.K. |last1=Guy |authorlink=Richard Guy |journal=Proof Techniques in Graph Theory (Ed. by F. Harary), Academic Press |volume= |year=1969 |pages=63–69 |ref=harv}}</ref>
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| The problem of determining the crossing number of the [[complete graph]] was first posed by [[Anthony Hill (artist)|Anthony Hill]], and appears in print in 1960.<ref name="nabla">{{Cite journal |title=A combinatorial problem |first1=R.K. |last1=Guy |authorlink=Richard Guy |journal=Nabla (Bulletin of the Malayan Mathematical Society) |volume=7 |year=1960 |pages=68–72 |ref=harv}}</ref> Hill and his collaborator [[John Ernest]] were two [[Constructivism (art)|constructionist artists]] fascinated by mathematics, who not only formulated this problem but also originated a conjectural upper bound for this crossing number, which [[Richard Guy]] published in 1960.,<ref name="nabla"/> namely
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| :<math>cr(K_p) \le (1/4) \left\lfloor\frac{p}{2}\right\rfloor\left\lfloor\frac{p-1}{2}\right\rfloor\left\lfloor\frac{p-2}{2}\right\rfloor\left\lfloor\frac{p-3}{2}\right\rfloor,</math>
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| which gives values of <math>1, 3, 9, 18, 36, 60, 100, 150 </math> for <math> p = 1, \ldots, 12; </math>
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| see sequence ({{OEIS link|A000241}}) in the OEIS.
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| An independent formulation of the conjecture was made by [[Thomas L. Saaty]] in 1964.<ref name="pnas">{{Cite journal |title=The minimum number of intersections in complete graphs |first1=T.L. |last1=Saaty |authorlink=Thomas L. Saaty |journal=Proceedings of the National Academy of Sciences of the United States of America|volume=52 |year=1964 |pages=688–690 |ref=harv}}</ref> Saaty further
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| verified that the upper bound is achieved for <math> p \leq 10 </math> and Pan and Richter showed that it also
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| is achieved for <math>p = 11, p = 12. </math> If only straight-line segments are permitted, then one needs more crossings. The rectilinear crossing numbers for ''K''<sub>''5''</sub> through ''K''<sub>''12''</sub> are 1, 3, 9, 19, 36, 62, 102, 153, ({{OEIS link|A014540}}) and values up to ''K''<sub>''27''</sub> are known, with ''K''<sub>''28''</sub> requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project.<ref>{{cite web | url = //www.ist.tugraz.at/staff/aichholzer/research/rp/triangulations/crossing/ | title = Rectilinear Crossing Number project | author = Oswin Aichholzer}}</ref> Interestingly, it is not known whether the ordinary and rectilinear crossing numbers are the same for bipartite complete graphs. If the Zarankiewicz conjecture is correct, then the formula for the crossing number of the complete graph is asymptotically correct;<ref name="jct">{{Cite journal |title=On a problem of P. Erdos |first1=P.C. |last1=Kainen |authorlink=Paul Chester Kainen |journal=Journal of Combinatorial Theory|volume=5 |year=1968 |pages=374–377 |ref=harv}}</ref> that is,
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| :<math> \lim_{p \to \infty} \; cr(K_p) \; 64/p^4 = 1. </math>
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| As of January 2012, crossing numbers are known for very few graph families. In particular, except for a few initial cases, the crossing number of complete graphs, bipartite complete graphs, and products of cycles all remain unknown. There has been some progress on lower bounds, as reported by {{harvtxt|de Klerk|Maharry|Pasechnik|Richter|2006}}.<ref>{{Cite journal |last1=de Klerk |first1=E. |last2=Maharry |first2=J. |last3=Pasechnik |first3=D. V. |last4=Richter |first4=B. |last5=Salazar |first5=G. |year=2006 |url=http://arno.uvt.nl/show.cgi?fid=71701 |title=Improved bounds for the crossing numbers of ''K<sub>m,n</sub>'' and ''K<sub>n</sub>'' |journal=[[SIAM Journal on Discrete Mathematics]] |volume=20 |issue=1 |pages=189–202 |doi=10.1137/S0895480104442741 |ref=harv |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref> | |
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| The [[Albertson conjecture]], formulated by Michael O. Albertson in 2007, states that, among all graphs with [[chromatic number]] ''n'', the complete graph ''K''<sub>''n''</sub> has the minimum number of crossings. That is, if the Guy-Saaty conjecture on the crossing number of the complete graph is valid, every ''n''-chromatic graph has crossing number at least equal to the formula in the conjecture. It is now known to hold for ''n'' ≤ 16.<ref>{{cite arXiv |first1=János |last1=Barát |first2=Géza |last2=Tóth |year=2009 |title=Towards the Albertson Conjecture |eprint=0909.0413 |ref=harv |class=math.CO}}</ref>
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| ==Complexity==
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| In general, determining the crossing number of a graph is hard; [[Michael Garey|Garey]] and [[David S. Johnson|Johnson]] showed in 1983 that it is an [[NP-hard]] problem.<ref>{{cite journal |author=[[Michael Garey|Garey, M. R.]]; [[David S. Johnson|Johnson, D. S.]] |title=Crossing number is NP-complete |journal=SIAM J. Alg. Discr. Meth. |volume=4 |pages=312–316 |year=1983 |mr=0711340 |doi=10.1137/0604033 |issue=3 |ref=harv}}</ref> In fact the problem remains NP-hard even when restricted to [[cubic graph]]s.<ref>{{cite journal |author=Hliněný, P. |title=Crossing number is hard for cubic graphs |year=2006 |journal=[[Journal of Combinatorial Theory|Journal of Combinatorial Theory, Series B]] |volume=96 |issue=4 |pages=455–471 |mr=2232384 |doi=10.1016/j.jctb.2005.09.009 |ref=harv}}</ref> More specifically, determining the rectilinear crossing number is [[Complete (complexity)|complete]] for the [[existential theory of the reals]].<ref>{{Cite conference|first=Marcus|last=Schaefer|title=Complexity of some geometric and topological problems|url=http://ovid.cs.depaul.edu/documents/convex.pdf|booktitle=[[International Symposium on Graph Drawing|Graph Drawing, 17th International Symposium, GS 2009, Chicago, IL, USA, September 2009, Revised Papers]]|series=Lecture Notes in Computer Science|publisher=Springer-Verlag|volume=5849|pages=334–344|doi=10.1007/978-3-642-11805-0_32|year=2010|ref=harv|isbn=978-3-642-11804-3}}.</ref>
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| On the positive side, there are efficient algorithms for determining if the crossing number is less than a fixed constant ''k'' — in other words, the problem is [[fixed-parameter tractable]].<ref>{{Cite journal |last=Grohe |first=M. |title=Computing crossing numbers in quadratic time |journal=J. Comput. System Sci. |volume=68 |issue=2 |year=2005 |pages=285–302 |mr=2059096 |doi=10.1016/j.jcss.2003.07.008 |postscript=<!--None--> |ref=harv}}; {{Cite conference |first1= Ken-ichi |last1=Kawarabayashi |first2=Bruce |last2=Reed | author2-link = Bruce Reed (mathematician) |title=Computing crossing number in linear time |booktitle=Proceedings of the 29th Annual ACM Symposium on Theory of Computing |year=2007 |pages=382–390 |doi=10.1145/1250790.1250848 |ref= harv |isbn= 978-1-59593-631-8}}</ref> It remains difficult for larger ''k'', such as |''V''|/2. There are also efficient [[approximation algorithm]]s for approximating cr(''G'') on graphs of bounded degree.<ref>{{Cite journal |first1=Guy |last1=Even |first2=Sudipto |last2=Guha |first3 = Baruch | last3 = Schieber | title=Improved Approximations of Crossings in Graph Drawings and VLSI Layout Areas |journal=[[SIAM Journal on Computing]] |volume=32 |year=2003 |pages=231–252 |doi=10.1137/S0097539700373520 |issue=1 |ref=harv}}</ref> In practice [[heuristic]] algorithms are used, such as the simple algorithm which starts with no edges and continually adds each new edge in a way that produces the fewest additional crossings possible. These algorithms are used in the Rectilinear Crossing Number<ref>[http://dist.ist.tugraz.at/cape5/ Rectilinear Crossing Number] on the Institute for Software Technology at Graz, University of Technology (2009).</ref> [[distributed computing]] project.
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| == Crossing numbers of cubic graphs ==
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| The smallest [[cubic graph]]s with crossing numbers 1–8 are known {{OEIS|id=A110507}}. The smallest 1-crossing cubic graph is the [[complete bipartite graph]] ''K''<sub>3,3</sub>, with 6 vertices. The smallest 2-crossing cubic graph is the [[Petersen graph]], with 10 vertices. The smallest 3-crossing cubic graph is the [[Heawood graph]], with 14 vertices. The smallest 4-crossing cubic graph is the [[Möbius-Kantor graph]], with 16 vertices. The smallest 5-crossing cubic graph is the [[Pappus graph]], with 18 vertices. The smallest 6-crossing cubic graph is the [[Desargues graph]], with 20 vertices. None of the four 7-crossing cubic graphs, with 22 vertices, are well known.<ref>{{MathWorld|urlname=GraphCrossingNumber|title=Graph Crossing Number}}</ref> The smallest 8-crossing cubic graph is the [[McGee graph]] or (3,7)-[[cage graph]], with 24 vertices.
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| In 2009, Exoo conjectured that the smallest cubic graph with crossing number 11 is the [[Coxeter graph]], the smallest cubic graph with crossing number 13 is the [[Tutte–Coxeter graph]] and the smallest cubic graph with crossing number 170 is the [[Tutte 12-cage]].<ref>{{cite web |last=Exoo |first=G. |url=http://isu.indstate.edu/ge/COMBIN/RECTILINEAR/ |title=Rectilinear Drawings of Famous Graphs}}</ref><ref>{{Cite journal |last1=Pegg |first1=E. T. |last2=Exoo |first2=G. |title=Crossing Number Graphs |journal=Mathematica Journal |volume=11 |year=2009 |ref=harv |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>
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| == The crossing number inequality ==
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| <!-- linked from redirect [[Crossing lemma]] -->
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| The very useful ''crossing number inequality'', discovered independently by [[Miklós Ajtai|Ajtai]], [[Václav Chvátal|Chvátal]], Newborn, and [[Endre Szemerédi|Szemerédi]]<ref>{{cite conference|author=Ajtai, M.; Chvátal, V.; Newborn, M.; Szemerédi, E.|title=Crossing-free subgraphs|booktitle=Theory and Practice of Combinatorics|series=North-Holland Mathematics Studies|volume=60|year=1982|pages=9–12|mr=0806962 }}</ref> and by Leighton,<ref>{{cite book |author=Leighton, T. |title=Complexity Issues in VLSI |series=Foundations of Computing Series |publisher=MIT Press |location=Cambridge, MA |year=1983}}</ref> asserts that if a graph ''G'' (undirected, with no loops or multiple edges) with ''n'' vertices and ''e'' edges satisfies
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| :<math>e > 7.5 n,\,</math>
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| then we have
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| :<math>\operatorname{cr}(G) \geq \frac{e^3}{33.75 n^2}.\,</math>
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| The constant 33.75 is the best known to date,
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| and is due to Pach and Tóth;<ref name="pt">{{cite journal |author = Pach, J.; Tóth, G. |title=Graphs drawn with few crossings per edge |journal=Combinatorica |volume=17 |year=1997 |pages=427–439 |mr=1606052 |doi=10.1007/BF01215922 |issue = 3 |ref = harv}}</ref> the constant 7.5 can be lowered to 4, but at the expense of replacing 33.75 with the worse constant of 64. | |
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| The motivation of Leighton in studying crossing numbers was for applications to [[VLSI]] design in theoretical | |
| computer science. Later, Székely<ref>{{cite journal |author=Székely, L. A. |title=Crossing numbers and hard Erdős problems in discrete geometry |journal=Combinatorics, Probability and Computing |volume=6 |year=1997 |pages=353–358 |mr=1464571 |doi=10.1017/S0963548397002976 |issue=3 |ref=harv}}</ref> also realized that this inequality yielded very simple proofs of some important theorems
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| in [[Incidence (geometry)|incidence geometry]], such as [[Beck's theorem (geometry)|Beck's theorem]] and the [[Szemerédi-Trotter theorem]], and Tamal Dey used it to prove upper bounds on [[K-set (geometry)|geometric ''k''-sets]].<ref>{{cite journal | author = Dey, T. L. | title = Improved bounds for planar ''k''-sets and related problems | journal = [[Discrete and Computational Geometry]] | volume = 19 | issue = 3 | year = 1998 | pages = 373–382 | doi = 10.1007/PL00009354 | mr = 1608878 | ref = harv}}</ref>
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| For graphs with [[Girth (graph theory)|girth]] larger than 2''r'' and ''e'' ≥ 4''n'', Pach, Spencer and Tóth<ref>{{cite journal | journal= [[Discrete and Computational Geometry]] | year=2000 | volume=24 | issue=4 | pages=623–644 | title=New bounds on crossing numbers | last1=Pach | first1=János | last2=Spencer | first2=Joel |last3=Tóth | first3=Géza | author1-link=János Pach | author2-link=Joel Spencer | ref= harv}}</ref>
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| demonstrated an improvement of this inequality to
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| :<math>\operatorname{cr}(G) \geq c_r\frac{e^{r+2}}{n^{r+1}}.\,</math>
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| === Proof of crossing number inequality ===
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| We first give a preliminary estimate: for any graph ''G'' with ''n'' vertices and ''e'' edges, we have
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| : <math>\operatorname{cr}(G) \geq e - 3n.\,</math>
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| To prove this, consider a diagram of ''G'' which has exactly cr(''G'') crossings. Each of these crossings can be removed
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| by removing an edge from ''G''. Thus we can find a graph with at least <math>e-\operatorname{cr}(G)</math> edges and ''n'' vertices
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| with no crossings, and is thus a [[planar graph]]. But from [[planar graph|Euler's formula]] we must then have
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| <math>e-\operatorname{cr}(G) \leq 3n</math>, and the claim follows. (In fact we have <math>e-\operatorname{cr}(G) \leq 3n-6</math> for ''n'' ≥ 3).
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| To obtain the actual crossing number inequality, we now use a [[probabilistic method|probabilistic argument]]. We let 0 < ''p'' < 1 be a [[probability]] parameter to be chosen later, and construct a [[random graph|random subgraph]] ''H'' of ''G'' by allowing each vertex of ''G'' to lie in ''H'' independently with probability ''p'', and allowing an edge of ''G'' to lie in ''H'' if and only if its two vertices were chosen to lie in ''H''. Let <math>e_H</math> denote the number of edges of ''H'', and let <math>n_H</math> denote the number of vertices.
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| Now consider a diagram of ''G'' with cr(''G'') crossings. We may assume that any two edges in this diagram with a common
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| vertex are disjoint, otherwise we could interchange the intersecting parts of the two edges and reduce the crossing number by one. Thus every crossing in this diagram involves four distinct vertices of ''G''.
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| Since ''H'' is a subgraph of ''G'', this diagram contains a diagram of ''H''; let <math>\operatorname{cr}_H</math> denote the number of crossings of this random graph. By the preliminary crossing number
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| inequality, we have
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| : <math> \operatorname{cr}_H \geq e_H - 3n_H.</math>
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| Taking [[expected value|expectation]]s we obtain
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| : <math>{\Bbb E}(\operatorname{cr}_H) \geq {\Bbb E(e_H)} - 3 {\Bbb E}(n_H).</math>
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| Since each of the ''n'' vertices in ''G'' had a probability ''p'' of being in ''H'', we have <math>{\Bbb E}(n_H) = pn</math>.
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| Similarly, since each of the edges in ''G'' has a probability <math>p^2</math> of remaining in ''H'' (since both endpoints
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| need to stay in ''H''), then
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| <math>{\Bbb E}(e_H) = p^2 e</math>. Finally, every crossing in the diagram of ''G'' has a probability <math>p^4</math> of remaining in ''H'', since every crossing involves four vertices, and so <math>{\Bbb E}(\operatorname{cr}_H) = p^4 \operatorname{cr}(G)</math>. Thus we have
| |
| | |
| : <math> p^4 \operatorname{cr}(G) \geq p^2 e - 3 p n.\,</math>
| |
| | |
| If we now set ''p'' to equal 4''n''/''e'' (which is less than one, since we assume that ''e'' is greater than 4''n''),
| |
| we obtain after some algebra
| |
| | |
| : <math> \operatorname{cr}(G) \geq e^3 / 64 n^2.\,</math>
| |
| | |
| A slight refinement of this argument allows one to replace 64 by 33.75 when ''e'' is greater than 7.5 ''n''.<ref name="pt"/>
| |
| | |
| ==See also==
| |
| *[[1-planar graph]]
| |
| | |
| == Notes ==
| |
| {{reflist|2}}
| |
| | |
| [[Category:Topological graph theory]]
| |
| [[Category:Inequalities]]
| |
| [[Category:Articles containing proofs]]
| |
| [[Category:Graph invariants]]
| |
| [[Category:Graph drawing]]
| |
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