On the Crossing Number of Kn without Computer Assistance

McQuillan, Dan; Richter, R. Bruce

doi:10.1002/jgt.21908

Cited by 8 publications

(11 citation statements)

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“…Guy's, or Harary and Hill's, conjecture states that cr(K n ) = Z(n) [274,70]; the conjecture is known to be true for n 12 [432], and cr(K 13 ) ∈ {219, 221, 223, 225} [385]. (For a computer-free proof that cr(K 9 ) = 36, see [386].) For a strengthened version of the conjecture, see [57].…”

Section: Crossing Numbermentioning

confidence: 99%

The Graph Crossing Number and its Variants: A Survey

Schaefer

2013

Electron. J. Combin.

100

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The crossing number is a popular tool in graph drawing and visualization, but there is not really just one crossing number; there is a large family of crossing number notions of which the crossing number is the best known. We survey the rich variety of crossing number variants that have been introduced in the literature for purposes that range from studying the theoretical underpinnings of the crossing number to crossing minimization for visualization problems.

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Section: Crossing Numbermentioning

confidence: 99%

The Graph Crossing Number and its Variants: A Survey

Schaefer

2013

Electron. J. Combin.

100

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“…Many results on crossing numbers of complete graphs and complete bipartite graphs rely crucially on observations about the responsibility of vertices in crossing‐minimal drawings. See, for instance, [1,4,6,8], and Sections 1.2 and 1.3 in [11].…”

Section: Discussionmentioning

confidence: 99%

Drawings of complete graphs in the projective plane

Arroyo

McQuillan

Richter

et al. 2021

Journal of Graph Theory

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Hill's Conjecture states that the crossing number cr ( K n ) of the complete graph K n in the plane (equivalently, the sphere) is 1 4 ⌊ n 2 ⌋ ⌊ n − 1 2 ⌋ ⌊ n − 2 2 ⌋ ⌊ n − 3 2 ⌋ = n 4 ∕ 64 + O ( n 3 ). Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely n 4 ∕ 64 + O ( n 3 ), thus matching asymptotically the conjectured value of cr ( K n ). Let cr P ( G ) denote the crossing number of a graph G in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of K n is ( n 4 ∕ 8 π 2 ) + O ( n 3 ). In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if lim n → ∞ cr P ( K n ) ∕ n 4 = 1 ∕ 8 π 2. We construct drawings of K n in the projective plane that disprove this.

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“…Our study of these drawings was motivated by a couple of specific events. One was the computer-free proof by two of the authors that the crossing number of K 9 is 36 [22]. As part of that proof, the two drawings K 3 5 and K 5 5 were both shown not to occur in any optimal (that is, fewest crossings) drawing of K 7 .…”

Section: Introductionmentioning

confidence: 99%

“…For a general drawing D of K n , we can ask whether there is a subdrawing D[K k ] such that one face is bounded by a k-cycle: this is a natural drawing of K k . (In [22], these drawings are quite appropriately labelled "convex". We think convex is very descriptive of the drawings considered in this work and expect there to be no confusion with the two quite different uses of the term "convex".)…”

Section: Introductionmentioning

confidence: 99%