Improved bounds for the crossing numbers of Km,n and K n
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It has been long conjectured that the crossing number ct(Km,n) of the complete bipartite graph Km,n equals the Zarankiewicz number Z(m, n) := [m-1/2] [m/2] [n-1/2] [n/2]. Another longstanding conjecture states that the crossing number cr(Kn) of the complete graph Kn equals Z(n) := 1/4[n/2] [n-1/2] [n-2/2] [n-3/3]. In this paper we show the following improved bounds on the asymptotic ratios of these crossing numbers and their conjectured values: (i) for each fixed m ≥ 9, lim n→∞ cr(Km,n)/Z(m, n) ≥ 0.83m/(m - 1); (ii) limn→∞ cr(Kn,n)/Z(n, n) ≥ 0.83; and (iii) limn→∞ cr(Kn)/Z(n) ≥ 0.83. The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8, respectively. These improved bounds are obtained as a consequence of the new bound cr(K7,n) ≥ 2.1796n2 -4.5n. To obtain this improved lower bound for cr(K 7,n), we use some elementary topological facts on drawings of K 2,7 to set up a quadratic program on 6! variables whose minimum p satisfies cr(K7, n) ≥ (p/2)n22 -4.5n, and then use state-of-the-art quadratic optimization techniques combined with a bit of invariant theory of permutation groups to show that p ≥ 4.3593. © 2006 Society for Industrial and Applied Mathematics.
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Copositive cone; Crossing number; Invariants and centralizer rings of permutation groups; Semidefinite programming Centralizer rings; Copositive cones; Crossing numbers; Permutation groups; Permutations; Semidefinite programming; Asymptotic analysis; Combinatorial mathematics; Number theory; Optimization; Quadratic programming; Set theory; Graph theory
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