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Around 1958, Hill conjectured that the crossing number CRg(K<sub>n</sub>) of the complete graph KK<sub>n</sub> is Z(n):=1/4 &#8970; n/2 &#8971; &#8970;(n-1)/2&#8971; &#8970; (n-2)/2 &#8971; &#8970; (n-3)/2 &#8971; and provided drawings of K<sub>n</sub> with exactly Z(n) crossings. Towards the end of the century, substantially different drawings of(More)
Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P 2 (G) = d 2 i denote the sum of the squares of the degrees, d 1 ,. .. , d v , of the vertices of G. It is known that the maximum value of P 2 (G) for G ∈ G(v, e) occurs at one or both of two special graphs in G(v, e)—the quasi-star graph or the quasi-complete graph. For each(More)
The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph K n in the plane is at least Z(n) := 1 4 n 2 n−1 2 n−2 2 n−3 2. In this paper, we settle the Harary-Hill conjecture for shellable drawings. We say that a drawing D of K n is s-shellable if there exist a subset S = {v 1 , v 2 ,. .. , v s } of the vertices and(More)
Let Mm,n(0, 1) denote the set of all m × n (0,1)-matrices and let In this paper we exhibit some new formulas for G(m, n) where n ≡ −1 (mod 4). Specifically, for m = nt + r where 0 ≤ r < n, we show that for all sufficiently large t, G(nt + r, n) is a polynomial in t of degree n that depends on the characteristic polynomial of the adjacency matrix of a(More)
Recently, Aichholzer, García, Orden, and Ramos derived a remarkably improved lower bound for the number of (≤ k)-edges in an n-point set, and as an immediate corollary an improved lower bound on the rectilinear crossing number of K n. We use simple allowable sequences to extend all their results to the more general setting of simple generalized(More)
Given a class C of geometric objects and a point set P , a C-matching of P is a set M = {C 1 ,. .. , C k } ⊆ C of elements of C such that each C i contains exactly two elements of P and each element of P lies in at most one C i. If all of the elements of P belong to some C i , M is called a perfect matching. If, in addition, all of the elements of M are(More)