## On the crossing numbers of complete graphs (2006)

Venue: | University of Waterloo |

Citations: | 2 - 1 self |

### BibTeX

@ARTICLE{Pan06onthe,

author = {Shengjun Pan},

title = {On the crossing numbers of complete graphs},

journal = {University of Waterloo},

year = {2006}

}

### OpenURL

### Abstract

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we prove two main results. The Triangle Conjecture asserts that the convex hull of any optimal rectilinear drawing of Kn must be a triangle (for n � 3). We prove that, for the larger class of pseudolinear drawings, the outer face must be a triangle. The other main result is the next step toward Guy’s Conjecture that the crossing number of Kn is 1

### Citations

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(Show Context)
Citation Context ..., 1)). The crossing number of a graph G, denoted as cr(G), is the minimum number of edge crossings over all drawings of G in the plane. A graph G is planar if and only if cr(G) = 0. Garey and Johnson =-=[15]-=- showed that determining the crossing number is an NP-complete problem. We also use cr(D) to denote the number of crossings in a drawing D. A rectilinear drawing , or straight line drawing, is a drawi... |

199 |
Graphs on Surfaces
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Citation Context ...ar graph of any optimal drawing of Kn, for n � 4, is 3-connected. It is well known that the dual graph of any simple and 3-connected graph is simple and 3-connected (for example, see Theorem 2.6.7 in =-=[27]-=-, page 46). Hence we have Corollary A.1.2. The dual graph of any good drawing of Kn, n � 4, is simple and 3-connected.sAppendix A. Connectivity 71 A.2 4-connectivity Let D be any good drawing of Kn, w... |

138 |
Graph theory, volume 173 of Graduate Texts in Mathematics
- Diestel
- 2005
(Show Context)
Citation Context ...phic if and only if G D 1 and G D 2 are isomorphic. Proof. By Corollary A.1.2 in Appendix A, GD1, GD2 are 3-connected, and obviously simple. According to Whitney’s Theorem (e.g., see Theorem 4.3.2 in =-=[14]-=-, page 96), if a planar graph is simple and 3-connected, it has a unique drawing up to isomorphism. Hence we only need to determine if G D 1, G D 2 are isomorphic. � Thus the problem of drawing isomor... |

109 | Crossing numbers and hard Erdős problems in discrete geometry
- Székely
- 1997
(Show Context)
Citation Context ... In [23] he used the crossing number to set lower bound for the VLSI layout area of the graph. Before this paper, the relevance of crossing number for engineering was well known already [11]. Székely =-=[31]-=- used Theorem 1.1.1 to give a new proof for the Szemerédi-Trotter theorem, which tells how many incidences can be among n points and m straight lines in the plane. In the same paper he also gave simpl... |

96 |
Complexity issues in VLSI
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(Show Context)
Citation Context ...d by Paul Turán [33]. The problem, mathematically, is to find the crossing number of the complete bipartite graph Km,n. For a general graph G, there is a lower bound on cr(G): Theorem 1.1.1 (Leighton =-=[23]-=-, Ajtai, Chvátal, Newborn and Szemerédi [6]). For any graph G with n vertices and e � cn edges, cr(G) � c − 3 c 3 e3 n2, where the constant factor (c − 3)/c 3 achieves its maximum at c = 4.5. More wor... |

71 |
Crossing-free subgraphs
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(Show Context)
Citation Context ...cally, is to find the crossing number of the complete bipartite graph Km,n. For a general graph G, there is a lower bound on cr(G): Theorem 1.1.1 (Leighton [23], Ajtai, Chvátal, Newborn and Szemerédi =-=[6]-=-). For any graph G with n vertices and e � cn edges, cr(G) � c − 3 c 3 e3 n2, where the constant factor (c − 3)/c 3 achieves its maximum at c = 4.5. More work in crossing numbers has focused on partic... |

54 | A combinatorial problem
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- 1967
(Show Context)
Citation Context ...rical drawing of K8 de Klerk, Maharry, Pasechnik, Richter and Salazar [12] showed that, for m � 9, cr(Km,n) lim n→∞ Z(m, n) � α m m − 1 , where α = 0.83. A recent paper [13] improved α to 0.8594. Guy =-=[18]-=- initiated the hunt for cr(Kn). He conjectured that: Conjecture 2 (Guy’s Conjecture). For any positive integer n, where Z(n) = 1 4 � n 2 He gave proofs for n � 10 in [19]. cr(Kn) = Z(n), �� � � �� � n... |

38 | A successful concept for measuring non-planarity of graphs: the crossing number
- Székely
(Show Context)
Citation Context ...l rectilinear drawing of K12 is shown in Figure 1.7 with only some of the edges drawn. The coordinates of the vertices are obtained from the web page [2]. 1.3 Applications of crossing numbers Székely =-=[32]-=- gave a very nice introduction to various applications of crossing numbers. When a graph is not planar, it is natural to ask “how far” a non-planar graph is x 10 4sChapter 1. Introduction 14 from bein... |

37 |
On the combinatorial classification of nondegenerate configurations in the plane
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(Show Context)
Citation Context ... pulled away from the other vertices as far away as we want, while the drawing remains isomorphic.sChapter 3 Circular Sequences The concept of circular sequences was introduced by Goodman and Pollack =-=[16]-=-. Circular sequences are combinatorial structures, which can be used to encode point sets in the plane. There is a close relationship between the number of “convex quadrilaterals” in circular sequence... |

36 |
On the crossing number of complete graphs
- Aichholzer, Aurenhammer, et al.
- 2002
(Show Context)
Citation Context ...ny n � 9, cr(Kn) = cr(Kn). Brodsky, Durocher and Gethner [9] proved that cr(K10) = 62, which is greater than cr(K10) = 60. The latest progress on cr(Kn) is made by Aichholzer, Aurenhammer and Krasser =-=[3, 4]-=-. By enumerating abstract order types on computers, they determined cr(Kn) for n up to 17. The rectilinear crossing number has a surprising connection with Sylvester’s fourpoint problem (Sylvester 186... |

31 |
Computing the K shortest paths: A new algorithm and an experimental comparison
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(Show Context)
Citation Context ...to find all the paths from v1 to S with length at most d + δ, δ ∈ {0, 1, 2}. One possible choice is to use the algorithm for searching for the k-shortest paths. There are several such algorithms, see =-=[20]-=- for example. We could input a large k, and stop as soon as we find the first path with length > d + δ. For convenience we didn’t use the k-shortest paths algorithm. In our code we used depth-first se... |

31 |
A note of welcome
- Turán
- 1977
(Show Context)
Citation Context ..., any rectilinear drawing is a good drawing.sChapter 1. Introduction 3 1.1 Crossing number The origin of crossing number is the “Turán’s brick factory problem” in 1944, first introduced by Paul Turán =-=[33]-=-. The problem, mathematically, is to find the crossing number of the complete bipartite graph Km,n. For a general graph G, there is a lower bound on cr(G): Theorem 1.1.1 (Leighton [23], Ajtai, Chvátal... |

30 | Finite graphs and networks: an introduction with applications - Busacker, Saaty - 1965 |

30 |
Schrijver A.: Reduction of symmetric semidefinite programs using the regular ∗-representation
- Klerk, Pasechnik
- 2007
(Show Context)
Citation Context ...oduction 4 Figure 1.2: A cylindrical drawing of K8 de Klerk, Maharry, Pasechnik, Richter and Salazar [12] showed that, for m � 9, cr(Km,n) lim n→∞ Z(m, n) � α m m − 1 , where α = 0.83. A recent paper =-=[13]-=- improved α to 0.8594. Guy [18] initiated the hunt for cr(Kn). He conjectured that: Conjecture 2 (Guy’s Conjecture). For any positive integer n, where Z(n) = 1 4 � n 2 He gave proofs for n � 10 in [19... |

26 | The crossing number of P - Richter, Salazar - 2003 |

26 |
Cyclic-order graphs and Zarankiewicz’s crossing-number conjecture
- Woodall
- 1993
(Show Context)
Citation Context ...d so implies that Z(m, n) is an upper bound for cr(Km,n). Zarankiewicz’s Conjecture has been verified, for min{m, n} � 6, by Kleitman [21] and, for the special cases 7 � m � 8, 7 � n � 10, by Woodall =-=[35]-=-.sChapter 1. Introduction 4 Figure 1.2: A cylindrical drawing of K8 de Klerk, Maharry, Pasechnik, Richter and Salazar [12] showed that, for m � 9, cr(Km,n) lim n→∞ Z(m, n) � α m m − 1 , where α = 0.83... |

24 |
The crossing number of K5,n
- Kleitman
- 1970
(Show Context)
Citation Context ...h vertex on the y-axis. This such drawing has Z(m, n) crossings and so implies that Z(m, n) is an upper bound for cr(Km,n). Zarankiewicz’s Conjecture has been verified, for min{m, n} � 6, by Kleitman =-=[21]-=- and, for the special cases 7 � m � 8, 7 � n � 10, by Woodall [35].sChapter 1. Introduction 4 Figure 1.2: A cylindrical drawing of K8 de Klerk, Maharry, Pasechnik, Richter and Salazar [12] showed that... |

23 |
Abstract order type extension and new results on the rectilinear crossing number
- Aichholzer, Krasser
(Show Context)
Citation Context ...ny n � 9, cr(Kn) = cr(Kn). Brodsky, Durocher and Gethner [9] proved that cr(K10) = 62, which is greater than cr(K10) = 60. The latest progress on cr(Kn) is made by Aichholzer, Aurenhammer and Krasser =-=[3, 4]-=-. By enumerating abstract order types on computers, they determined cr(Kn) for n up to 17. The rectilinear crossing number has a surprising connection with Sylvester’s fourpoint problem (Sylvester 186... |

22 |
Relations between crossing numbers of complete and complete bipartite graphs
- Richter, Thomassen
- 1997
(Show Context)
Citation Context ...that: Conjecture 1 (Zarankiewicz’s Conjecture). For any positive integers m and n, where Z(m, n) = cr(Km,n) = Z(m, n), � m � 2 � � m − 1 �n� 2 2 � � n − 1 . 2 As described by Richter and Thomassen in =-=[28]-=-, a drawing of Km,n with Z(m, n) crossings can be obtained as follows: place m vertices along the x-axis, with ⌊m/2⌋ on the positive half and ⌈m/2⌉ on the negative half. Place n vertices along the y-a... |

21 | Toward the rectilinear crossing number of Kn: new drawings, upper bounds, and asymptotics
- Brodsky, Durocher, et al.
(Show Context)
Citation Context ...decreasing and is at most 1. Therefore 4 lim n→∞ cr(Kn) � exists. Thus the recursive construction implies that lim n→∞ � n 4 n→∞ � n 4 cr(Kn) cr(Dn) � � lim � ≈ 0.3846 . Brodsky, Durocher and Gethner =-=[10]-=- improved the recursive construction by sliding the three copies of flattened Dn/3, as shown Figure 1.4, where a is defined in terms of n, and represents the number of vertices that still dock on the ... |

21 | The rectilinear crossing number of a complete graph and Sylvester’s ”Four Point” problem of geometric probability
- Scheinerman, E, et al.
- 1994
(Show Context)
Citation Context ...ility q(R) that four points which are chosen independently uniformly at random in R form a convex quadrilateral. Let q∗ = infR q(R). In 1994, more than 100 years after Sylvester, Scheinerman and Wilf =-=[29]-=- proved that q∗ = lim n→∞ cr(Kn) � . 1.2.1 Constructions for upper bounds To date, searching for better upper bounds for cr(Kn) has been based on constructions of rectilinear drawings with few crossin... |

21 |
On a problem of P. Turán concerning graphs
- Zarankiewicz
- 1954
(Show Context)
Citation Context ...onstant factor (c − 3)/c 3 achieves its maximum at c = 4.5. More work in crossing numbers has focused on particular graphs. We will introduce the progress on cr(Km,n), cr(Kn) and cr(Kn). Zarankiewicz =-=[36]-=- conjectured that: Conjecture 1 (Zarankiewicz’s Conjecture). For any positive integers m and n, where Z(m, n) = cr(Km,n) = Z(m, n), � m � 2 � � m − 1 �n� 2 2 � � n − 1 . 2 As described by Richter and ... |

19 |
Crossing number of graphs, in
- Guy
- 1972
(Show Context)
Citation Context ...13] improved α to 0.8594. Guy [18] initiated the hunt for cr(Kn). He conjectured that: Conjecture 2 (Guy’s Conjecture). For any positive integer n, where Z(n) = 1 4 � n 2 He gave proofs for n � 10 in =-=[19]-=-. cr(Kn) = Z(n), �� � � �� � n − 1 n − 2 n − 3 . 2 2 2 Richter and Thomassen [28] gave a detailed description of a cylindrical drawing of Kn, for each even n, with exactly Z(n) crossings. The drawing ... |

17 | Convex quadrilaterals and k-sets
- Lovász, Vesztergombi, et al.
(Show Context)
Citation Context ...tly two adjacent positions. More details of the connection between cr(Kn) and circular sequences are explained in Chapter 3. By manipulating circular sequences, Lovász, Vesztergombi, Wagner and Welzl =-=[24]-=- proved that Since cr(Kn) � lim n→∞ � � � � 3 n + 10−5 8 4 cr(Kn) � = 3 8 lim � n 4 n→∞ cr(Kn) Z(n) + O(n 3 ). � 3 8 , the asymptotic lower bound Lovász et al. gave implies that, for n large enough, c... |

10 | k-sets, convex quadrilaterals, and the rectilinear crossing number
- Balogh, Salazar
(Show Context)
Citation Context ...et al. gave implies that, for n large enough, cr(Kn) is different from cr(Kn). Based on the same idea of circular sequences, by modeling and solving a digraph optimization problem, Balogh and Salazar =-=[8]-=- improved the asymptotic lower bound to � � n 0.37553 + O(n 4 3 ), which is the largest lower bound known to date. Balogh, Leaños and Salazar have shown that, for n ≥ 10, cr(Kn) > cr(Kn) (personal com... |

10 |
A note on the parity of the number of crossings of a graph
- Kleitman
- 1976
(Show Context)
Citation Context ...∗ 8 19 27 9 36 46 10 62 74 v1 v4 n cr(Kn) cr(Dn,�4) 11 102 114 12 153 168 13 229 241 14 324 335 15 447 455 16 603 606 17 798 792 Table 2.1: Lower bounds for cr(Dn, �4) ∗ Due to the parity argument in =-=[22]-=- by Kleitman, for each odd n, the numbers of crossings in any good drawings of Kn have the same parity.sChapter 2. Warm Up 26 n non-isomorphic drawings 3 1 3 4 1 3, 1 5 1 3, 2 6 1 3, 3 7 3 3, 3, 1 h1(... |

7 |
On the structure of sets minimizing the rectilinear crossing number
- Aichholzer, Orden, et al.
- 2006
(Show Context)
Citation Context .... Although our proof on the pseudolinearsChapter 3. Circular Sequences 46 drawings does not imply the Triangle Conjecture, the recently announced proof of the Triangle Conjecture by Aichholzer et al. =-=[5]-=- does not seem to imply our result either. 3.4 Extreme vertices of a drawing Let ej be the number of j-switches in a circular sequence Π on n elements, and Ej be the number of (� j)-switches, i.e. Ej ... |

5 |
The rectilinear crossing number of certain graphs
- Singer
- 1971
(Show Context)
Citation Context ...nstructions for upper bounds To date, searching for better upper bounds for cr(Kn) has been based on constructions of rectilinear drawings with few crossings. Singer suggests a recursive construction =-=[30, 34]-=- of rectilinear drawings Dn of Kn, where n = 3 k . For k = 1, draw Kn as a triangle. To construct Dn, n = 3 k , we first flatten the constructed drawing Dn/3, so � n 4sChapter 1. Introduction 8 Figure... |

4 |
The nauty page. http://cs.anu.edu.au/ ∼ bdm/ nauty
- McKay
- 2004
(Show Context)
Citation Context ...r determining the automorphism group of a vertex-colored graph. It is also able to produce a canonically labelled isomorph of the graph, which can be used to assist in isomorphism testing. Please see =-=[26]-=- for more details. 67sAppendix A Connectivity We prove that, for any n � 4, the planar graph of any good drawing of Kn is 3connected, which has an application in the proof of Algorithm 1 for generatin... |

3 |
On crossing numbers, and some unsolved problems
- Wilf
- 1997
(Show Context)
Citation Context ...nstructions for upper bounds To date, searching for better upper bounds for cr(Kn) has been based on constructions of rectilinear drawings with few crossings. Singer suggests a recursive construction =-=[30, 34]-=- of rectilinear drawings Dn of Kn, where n = 3 k . For k = 1, draw Kn as a triangle. To construct Dn, n = 3 k , we first flatten the constructed drawing Dn/3, so � n 4sChapter 1. Introduction 8 Figure... |

2 |
The rectilinear crossing number of K10 is 62. eprint arXiv:cs.DM/0009023
- Brodsky, Durocher, et al.
- 2000
(Show Context)
Citation Context ...d formula or conjectured optimal drawings for cr(Kn). Guy [19] gave cr(Kn) for n � 9. Except for n = 8 where cr(K8) = 19 and cr(K8) = 18, for any n � 9, cr(Kn) = cr(Kn). Brodsky, Durocher and Gethner =-=[9]-=- proved that cr(K10) = 62, which is greater than cr(K10) = 60. The latest progress on cr(Kn) is made by Aichholzer, Aurenhammer and Krasser [3, 4]. By enumerating abstract order types on computers, th... |

1 |
Crossing Number Examples. http://www.ist.tugraz.at/staff/aichholzer/crossings.html
- Aichholzer
(Show Context)
Citation Context ...f any such drawing is indeed a triangle. An optimal rectilinear drawing of K12 is shown in Figure 1.7 with only some of the edges drawn. The coordinates of the vertices are obtained from the web page =-=[2]-=-. 1.3 Applications of crossing numbers Székely [32] gave a very nice introduction to various applications of crossing numbers. When a graph is not planar, it is natural to ask “how far” a non-planar g... |

1 |
Improved bounds for the crossing numbers of Km,n and Kn. eprint arXiv:math/0404142
- Klerk, Maharry, et al.
- 2004
(Show Context)
Citation Context ...by Kleitman [21] and, for the special cases 7 � m � 8, 7 � n � 10, by Woodall [35].sChapter 1. Introduction 4 Figure 1.2: A cylindrical drawing of K8 de Klerk, Maharry, Pasechnik, Richter and Salazar =-=[12]-=- showed that, for m � 9, cr(Km,n) lim n→∞ Z(m, n) � α m m − 1 , where α = 0.83. A recent paper [13] improved α to 0.8594. Guy [18] initiated the hunt for cr(Kn). He conjectured that: Conjecture 2 (Guy... |

1 |
Semispaces of confgurations, cell complexes of arrangements
- Goodman, Pollack
- 1984
(Show Context)
Citation Context ...3.3.1 the sequences constructed with passing extremity operation may not be geometrically realizable. Thus Theorem 3.3.2 does NOT lead to a proof of the Triangle Conjecture. However, in another paper =-=[17]-=- Goodman and Pollack proved that every circular sequence is realizable by a pseudolinear drawing. A pseudoline is a simple closed curve in the projective plane P 2 which does not disconnect P 2 . Give... |