## New Bounds on Crossing Numbers (1999)

Citations: | 12 - 4 self |

### BibTeX

@MISC{Pach99newbounds,

author = {Janos Pach and Joel Spencer and Géza Tóth},

title = {New Bounds on Crossing Numbers},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by #(n, e) the minimum of cr(G) taken over all graphs with n vertices and at least e edges. We prove a conjecture of P. Erdos and R. Guy by showing that #(n, e)n 2 /e 3 tends to a positive constant as n ## and n # e # n 2 . Similar results hold for graph drawings on any other surface of fixed genus. We prove better bounds for graphs satisfying some monotone properties. In particular, we show that if G is a graph with n vertices and e # 4n edges, which does not contain a cycle of length four (resp. six), then its crossing number is at least ce 4 /n 3 (resp. ce 5 /n 4 ), where c > 0 is a suitable constant. These results cannot be improved, apart from the value of the constant. This settles a question of M. Simonovits. 1 Introduction Let G be a simple undirected graph with n(G) nodes (vertices) and e(G) edges. A drawing of G in the plane is a m...

### Citations

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Citation Context ...Leighton [L83] observed that there is an intimate relationship between the bisection width and the crossing number of a graph, which is based on the Lipton--Tarjan separator theorem for planar graphs =-=[LT79]-=-. The proofs of Theorems 1-3 are based on repeated application of the following version of this relationship. Theorem B. [PSS96] Let G be a graph of n vertices, whose degrees are d 1 , d 2 , . . . , d... |

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Citation Context ...at no three edges have an interior point in common. The crossing number, cr(G), of G is the minimum number of crossing points in any drawing of G. The determination of cr(G) is an NP-complete problem =-=[GJ83]-=-. It was discovered by Leighton [L84] that the crossing number can be used to estimate the chip area required for the VLSI circuit # Supported by NSF grant CCR-94-24398 and PSC-CUNY Research Award 667... |

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Citation Context ...h is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) = O(n 1+1/r ). For r = 2, 3 and 5, this bound is tight. Claim C 00 . =-=[KST54]-=-, [F96], [ER62], [B66], [ARS98] For any integers s # r # 2, the maximum number of edges of a K r,s -free graph of n vertices, satisfies ex(n, K r,s ) = O(n 2-1/r ). 9 This bound is tight for s > (r - ... |

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(Show Context)
Citation Context ...des and e(G) = e edges, e # 4n. Then we have cr(G) # 1 33.75 e 3 n 2 . Theorem A can be used to deduce the best known upper bounds for the number of unit distances determined by n points in the plane =-=[S98]-=-, for the number of di#erent ways how a line can split a set of n points into two equal parts [D98], and it has some other interesting corollaries [PS98]. It is easy to see that the bound in Theorem A... |

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Citation Context ...uce the best known upper bounds for the number of unit distances determined by n points in the plane [S98], for the number of di#erent ways how a line can split a set of n points into two equal parts =-=[D98]-=-, and it has some other interesting corollaries [PS98]. It is easy to see that the bound in Theorem A is tight, apart from the value of the constant. However, as it was suggested by Miklos Simonovits ... |

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Citation Context ...nt in common. The crossing number, cr(G), of G is the minimum number of crossing points in any drawing of G. The determination of cr(G) is an NP-complete problem [GJ83]. It was discovered by Leighton =-=[L84]-=- that the crossing number can be used to estimate the chip area required for the VLSI circuit # Supported by NSF grant CCR-94-24398 and PSC-CUNY Research Award 667339. + Supported by DIMACS Center, OT... |

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Citation Context ...It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . =-=[BS74]-=-, [B66], [Be66], [S66], [W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, whi... |

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Citation Context ...at is, #(n, e) = min n(G) = n e(G) = e cr(G). It follows from Theorem A that, for e # 4n, #(n, e)n 2 /e 3 is bounded from below and from above by two positive constants. Paul Erdos and Richard K. Guy =-=[EG73]-=- conjectured that if e # n then lim #(n, e)n 2 /e 3 exists. (We use the notation f(n) # g(n) to express that lim n## f(n)/g(n) = #.) In Section 4, we settle this problem. Theorem 4. If n # e # n 2 , t... |

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Citation Context ...r each 1 # i # n. Consequently, w(V (H)) = n. Since H is drawn on S g without crossing, H does not contain K # as a minor, where # = #4 + 4 # g# [RY68]. Then, by a result of Alon, Seymour, and Thomas =-=[AST90]-=-, the vertices of H can be partitioned into three sets, A, B and C, such that w(A), w(B) # n/3 and |C| # 25(1 + g 3/4 ) # cr g (G) + # n i=1 d 2 i , and there is no edge from A to B. Let A i = A#V i ,... |

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Citation Context ...ough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . [BS74], =-=[B66]-=-, [Be66], [S66], [W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has ... |

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Citation Context ... notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . [BS74], [B66], =-=[Be66]-=-, [S66], [W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property... |

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Citation Context ...aximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) = O(n 1+1/r ). For r = 2, 3 and 5, this bound is tight. Claim C 00 . [KST54], [F96], [ER62], [B66], =-=[ARS98]-=- For any integers s # r # 2, the maximum number of edges of a K r,s -free graph of n vertices, satisfies ex(n, K r,s ) = O(n 2-1/r ). 9 This bound is tight for s > (r - 1)!. In case r = 3, we obtain t... |

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Citation Context ...ices belonging to #. With this notation, w(V i ) = 1 for each 1 # i # n. Consequently, w(V (H)) = n. Since H is drawn on S g without crossing, H does not contain K # as a minor, where # = #4 + 4 # g# =-=[RY68]-=-. Then, by a result of Alon, Seymour, and Thomas [AST90], the vertices of H can be partitioned into three sets, A, B and C, such that w(A), w(B) # n/3 and |C| # 25(1 + g 3/4 ) # cr g (G) + # n i=1 d 2... |

33 |
On a problem in the theory of graphs
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(Show Context)
Citation Context ... 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) = O(n 1+1/r ). For r = 2, 3 and 5, this bound is tight. Claim C 00 . [KST54], [F96], =-=[ER62]-=-, [B66], [ARS98] For any integers s # r # 2, the maximum number of edges of a K r,s -free graph of n vertices, satisfies ex(n, K r,s ) = O(n 2-1/r ). 9 This bound is tight for s > (r - 1)!. In case r ... |

32 |
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(Show Context)
Citation Context ...owing general lower bound for cr(G), which was discovered independently by Ajtai, Chvatal, Newborn, and Szemeredi. The best known constant, 1/33.75, in the theorem is due to Pach and Toth. Theorem A. =-=[ACNS82]-=-, [L84], [PT97] Let G be a graph with n(G) = n nodes and e(G) = e edges, e # 4n. Then we have cr(G) # 1 33.75 e 3 n 2 . Theorem A can be used to deduce the best known upper bounds for the number of un... |

31 | On the number of incidences between points and curves
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- 1998
(Show Context)
Citation Context ...t distances determined by n points in the plane [S98], for the number of di#erent ways how a line can split a set of n points into two equal parts [D98], and it has some other interesting corollaries =-=[PS98]-=-. It is easy to see that the bound in Theorem A is tight, apart from the value of the constant. However, as it was suggested by Miklos Simonovits [S97], it may be possible to strengthen the theorem fo... |

28 |
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- 1991
(Show Context)
Citation Context ...litting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . [BS74], [B66], [Be66], [S66], =-=[W91]-=- For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisf... |

27 | Planar separators - Alon, Seymour, et al. - 1994 |

27 |
New asymptotics for bipartite Turán numbers
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(Show Context)
Citation Context ...er than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) = O(n 1+1/r ). For r = 2, 3 and 5, this bound is tight. Claim C 00 . [KST54], =-=[F96]-=-, [ER62], [B66], [ARS98] For any integers s # r # 2, the maximum number of edges of a K r,s -free graph of n vertices, satisfies ex(n, K r,s ) = O(n 2-1/r ). 9 This bound is tight for s > (r - 1)!. In... |

26 |
Über ein problem von K
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- 1958
(Show Context)
Citation Context ...resulting graph, e(G k ) # e 2 . On the other hand, each component of G k has relatively few vertices: n(G k j )s(2/3) k n(G # )se 2 16n 2 (G # ) = e 2 16n 2 (G k ) (j = 1, 2, . . . , M k ). Claim C. =-=[R58]-=- Let ex(n, K 2,2 ) denote the maximum number of edges that a K 2,2 -free graph with n vertices can have. Then ex(n, K 2,2 ) # n # 1 + # 4n - 3 # 4 # n 3/2 . Applying the Claim to each G j k , we obtai... |

25 |
Applications of the crossing number. Algorithmica 16
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- 1996
(Show Context)
Citation Context ...which is based on the Lipton--Tarjan separator theorem for planar graphs [LT79]. The proofs of Theorems 1-3 are based on repeated application of the following version of this relationship. Theorem B. =-=[PSS96]-=- Let G be a graph of n vertices, whose degrees are d 1 , d 2 , . . . , d n . Then b(G) # 10 # cr(G) + 2 # # # # n # i=1 d 2 i . Let #(n, e) denote the minimum crossing number of a graph G with n verti... |

24 |
Graphs drawn with few crossings per edge, Combinatorica 17:3
- Pach, Tóth
- 1997
(Show Context)
Citation Context ...er bound for cr(G), which was discovered independently by Ajtai, Chvatal, Newborn, and Szemeredi. The best known constant, 1/33.75, in the theorem is due to Pach and Toth. Theorem A. [ACNS82], [L84], =-=[PT97]-=- Let G be a graph with n(G) = n nodes and e(G) = e edges, e # 4n. Then we have cr(G) # 1 33.75 e 3 n 2 . Theorem A can be used to deduce the best known upper bounds for the number of unit distances de... |

23 | Graphs drawn with a few crossings per edge - Pach, Tóth - 1997 |

22 |
Relations between crossing numbers of complete and complete bipartite graphs
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- 1997
(Show Context)
Citation Context ... C # .029. 2. It follows from the convexity of the function # that the condition e # n in Theorem 4 is necessary. 3. Theorem 4 does not remain true if we drop the condition e # n 2 . It is known (see =-=[RT97]-=-) that cr(K n ) > (C + #) e 3 n 2 for a suitable positive constant # (where e = # n 2 # ). However, we were unable to decide whether e # n 2 can be replaced by a weaker assumption. 5 Midrange crossing... |

14 | Beineke, Topological Graph Theory, Selected Topics in Graph Theory - White, W - 1978 |

11 | Crossing numbers of graphs, in: Graph theory and applications - Guy - 1972 |

10 |
On minimal graphs of maximum even girth
- Singleton
- 1966
(Show Context)
Citation Context ...that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . [BS74], [B66], [Be66], =-=[S66]-=-, [W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r ,... |

1 |
personal communication
- Simonovits
(Show Context)
Citation Context ..., and it has some other interesting corollaries [PS98]. It is easy to see that the bound in Theorem A is tight, apart from the value of the constant. However, as it was suggested by Miklos Simonovits =-=[S97]-=-, it may be possible to strengthen the theorem for some special classes of graphs, e.g., for graphs not containing some fixed, so-called forbidden subgraph. In Sections 2 and 3 of the present paper we... |