Silvia Fernández-Merchant

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Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P2(G) = ∑ d i denote the sum of the squares of the degrees, d1, . . . , dv, of the vertices of G. It is known that the maximum value of P2(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 pair(More)
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)
Given a class C of geometric objects and a point set P , a C-matching of P is a set M = {C1, . . . , Ck} ⊆ C of elements of C such that each Ci contains exactly two elements of P and each element of P lies in at most one Ci. If all of the elements of P belong to some Ci, M is called a perfect matching. If, in addition, all of the elements of M are pairwise(More)
Even the most super cial glance at the vast majority of crossing-minimal geometric drawings of Kn reveals two hard-to-miss features. First, all such drawings appear to be 3-fold symmetric (or simply 3-symmetric). And second, they all are 3-decomposable, that is, there is a triangle T enclosing the drawing, and a balanced partition A,B, C of the underlying(More)
The Harary-Hill Conjecture states that the number of crossings in any drawing of the complete graph Kn 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 Kn is s-shellable if there exist a subset S = {v1, v2, . . . , vs} of the(More)