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The 2page crossing number of Kn
, 2011
"... Around 1958, Hill conjectured that the crossing number cr(Kn) of the complete graph Kn is Z (n): = 1 n n−1 n−2 n−3 4 2 2 2 2 and 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 ..."
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Around 1958, Hill conjectured that the crossing number cr(Kn) of the complete graph Kn is Z (n): = 1 n n−1 n−2 n−3 4 2 2 2 2 and 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 2page book drawings, that is, drawings where all the vertices are on a line ℓ (the spine) and each edge is fully contained in one of the two halfplanes (pages) defined by ℓ. The2page crossing number of Kn, denoted by ν2(Kn), is the minimum number of crossings determined by a 2page book drawing of Kn. It was generally conjectured that cr(Kn) =Z(n) and since cr(Kn) ≤ ν2(Kn) ≤ Z(n), the conjecture ν2(Kn) =Z(n) appeared as a milestone in the way to find the correct values of cr(Kn). 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 kedges for finite sets of points in the plane to topological drawings of Kn. We also introduce the concept of ≤≤kedges 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 kedges to the topological setting. 1 1
unknown title
, 2012
"... We recall that a book with k pages consists of a straight line (the spine) and k halfplanes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is ..."
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We recall that a book with k pages consists of a straight line (the spine) and k halfplanes (the pages), such that the boundary of each page is the spine. If a graph is drawn on a book with k pages in such a way that the vertices lie on the spine, and each edge is contained in a page, the result is a kpage book drawing (or simply a kpage drawing). The pagenumber of a graph G is the minimum k such that G admits a kpage embedding (that is, a kpage drawing with no edge crossings). The kpage crossing number νk(G) of G is the minimum number of crossings in a kpage drawing of G. We investigate the pagenumbers and kpage crossing numbers of complete bipartite graphs. We find the exact pagenumbers of several complete bipartite graphs, and use these pagenumbers to find the exact kpage crossing number of Kk+1,n for k ∈ {3, 4, 5, 6}. We also prove the general asymptotic estimate limk→ ∞ limn→ ∞ νk(Kk+1,n)/(2n2 /k2) = 1. Finally, we give general upper bounds for νk(Km,n), and relate these bounds to the kplanar
The Graph Crossing Number and its Variants: A Survey
"... 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 introdu ..."
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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. 1 So, Which Crossing Number is it? The crossing number, cr(G), of a graph G is the smallest number of crossings required in any drawing of G. Or is it? According to a popular introductory textbook on combinatorics [320, page 40] the crossing number of a graph is “the minimum number of pairs of crossing edges in a depiction of G”. So, which one is it? Is there even a difference? To start with the second question, the easy answer is: yes, obviously there is a difference, the difference between counting all crossings and counting pairs of edges that cross. But maybe these different ways of counting don’t make a difference and always come out