Special numbers of crossings for complete graphs

Harborth, Heiko

doi:10.1016/s0012-365x(01)00078-4

Cited by 17 publications

(33 citation statements)

References 3 publications

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“…Let us begin with proofs of the strong version. Two papers in 1976, one by Kleitman [8], the other by Harborth [6] showed that the parity of iocr(G) is independent of the drawing of G if G is either K 2j +1 or K 2j +1,2j +1 . Norine [9] supplies a different proof of this result and observes that it implies the strong version of the Hanani-Tutte theorem by an application of Kuratowski's theorem.…”

Section: Remarkmentioning

confidence: 99%

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Removing even crossings

Pelsmajer

Schaefer

Štefankovič

2007

Journal of Combinatorial Theory, Series B

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An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn so that its even edges are not involved in any intersections. We give a new and significantly simpler proof of the stronger statement that the redrawing can be done in such a way that no new odd intersections are introduced. We include two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowski's theorem), and the new result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.

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Section: Remarkmentioning

confidence: 99%

“…Thus, if we pick three edges that form no odd pairs, we may assume that they form no crossings at all. Sequentially add the other three edges; in order to cross only its partner an odd number of times, the rotations must be equivalent to (1,2,3,4,5,6), (2,1,4,3,6,5), which can be drawn with exactly 3 crossings. 2…”

Section: Small Crossing Numbersmentioning

confidence: 99%

Removing even crossings

Pelsmajer

Schaefer

Štefankovič

2007

Journal of Combinatorial Theory, Series B

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“…4 There are several proofs of this theorem [4,14,5,6,13,7] starting with the original papers by Hanani and Tutte. Kleitman's proof [6] is particularly short and elegant.…”

Section: The Weak Hanani-tutte Theorem On a Surfacementioning

confidence: 99%

“…This approach seems hopeless for surfaces other than the plane (the list of excluded minors is not even known yet for the torus). 5 In [11,12] we gave a new proof of the Hanani-Tutte theorem in the plane which avoids Kuratowski's theorem and uses elementary topological methods only.…”

Section: The Weak Hanani-tutte Theorem On a Surfacementioning

confidence: 99%

Removing Even Crossings on Surfaces

Pelsmajer

Schaefer

Štefankovič

2007

Electronic Notes in Discrete Mathematics

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We give a new, topological proof that the weak Hanani-Tutte theorem is true on orientable surfaces and extend the result to nonorientable surfaces. That is, we show that if a graph G cannot be embedded on a surface S, then any drawing of G on S must contain two edges that cross an odd number of times. We apply the result and proof techniques to obtain new and old results about generalized thrackles, including that every bipartite generalized thrackle in a surface S can be embedded in S. We also extend to arbitrary surfaces a result of Pach and Tóth that allows the redrawing of a graph so as to remove all crossings with even edges (an edge is even if it crosses every other edge an even number of times). From this result we can conclude that cr S (G), the crossing number of a graph G on surface S, is bounded by 2 ocr S (G) 2 , where ocr S (G) is the odd crossing number of G on surface S. Finally, we show that ocr S (G) = cr S (G) whenever ocr S (G) ≤ 2, for any surface S.

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“…The 2-page crossing number ν2(G) of a graph G is the minimum of cr(D) taken over all 2-page book drawings D of G. Alternative terminologies for the 2-page crossing number are circular crossing number [16] and fixed linear crossing number [7]. We may regard the pages as the closed half-planes defined by the spine, and so every 2-page book drawing can be realized as a plane drawing; it follows that cr(G) ≤ ν2(G) for every graph G.…”

Section: Introductionmentioning

confidence: 99%

The 2-page crossing number of K _n

Ábrego

Aichholzer

Fernández-Merchant

et al. 2012

Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry

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Around 1958, Hill conjectured that the crossing number cr(Kn) of the complete graph Kn is Z (n) := 1 4 n 2 n − 1 2 n − 2 2 n − 3 2and provided drawings of Kn with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of Kn with Z(n) crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by . The 2-page crossing number of Kn, denoted by ν2(Kn), is the minimum number of crossings determined by a 2-page book drawing of Kn. Since cr(Kn) ≤ ν2(Kn) and ν2(Kn) ≤ Z(n), a natural step towards Hill's Conjecture is the weaker conjecture ν2(Kn) = Z(n), that was popularized by Vrt'o. In this paper we develop a novel and innovative technique to investigate crossings in drawings of Kn, and use it to prove that ν2(Kn) = Z(n). To this end, we extend the inherent geometric definition of k-edges for finite sets of points in the plane to topological drawings of Kn. We also introduce the concept of ≤≤k-edges as a useful generalization of ≤kedges. Finally, we extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of Kn in terms of its number of k-edges to the topological setting.

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Special numbers of crossings for complete graphs

Cited by 17 publications

References 3 publications

Removing even crossings

Removing even crossings

Removing Even Crossings on Surfaces

The 2-page crossing number of K _n

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Special numbers of crossings for complete graphs

Cited by 17 publications

References 3 publications

Removing even crossings

Removing even crossings

Removing Even Crossings on Surfaces

The 2-page crossing number of K n

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The 2-page crossing number of K _n