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- W3103545947 abstract "It has long been conjectured that the crossing numbers of the complete bipartite graph $K_{m,n}$ and of the complete graph $K_n$ equal $Z(m,n):=bigllfloorfrac{n}{2}bigrrfloor bigllfloorfrac{n-1}{2}bigrrfloor bigllfloorfrac{m}{2}bigrrfloor bigllfloorfrac{m-1}{2}bigrrfloor$ and $Z(n):=frac{1}{4} bigllfloorfrac{n}{2}bigrrfloor bigllfloorfrac{n-1}{2}bigrrfloor bigllfloorfrac{n-2}{2}bigrrfloor bigllfloorfrac{n-3}{2}bigrrfloor$, respectively. In a $2$-page drawing of a graph, the vertices are drawn on a straight line (the spine), and each edge is contained in one of the half-planes of the spine. The $2$-page crossing number $nu_2(G)$ of a graph $G$ is the minimum number of crossings in a $2$-page drawing of $G$. Somewhat surprisingly, there are $2$-page drawings of $K_{m,n}$ (respectively, $K_n$) with exactly $Z(m,n)$ (respectively, $Z(n)$) crossings, thus yielding the conjectures (I) $nu_2(K_{m,n}) stackrel{?}{=} Z(m,n)$ and (II) $nu_2(K_n) stackrel{?}{=} Z(n)$. It is known that (I) holds for $min{m,n} le 6$, and that (II) holds for $n le 14$. In this paper we prove that (I) holds asymptotically (that is, $lim_{ntoinfty} nu_2(K_{m,n})/Z(m,n) =1$) for $m=7$ and $8$. We also prove (II) for $15 le n le 18$ and $n in {20,24}$, and establish the asymptotic estimate $lim_{ntoinfty} nu_2(K_{n})/Z(n) ge 0.9253.$ The previous best-known lower bound involved the constant $0.8594$." @default.
- W3103545947 created "2020-11-23" @default.
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- W3103545947 date "2012-01-01" @default.
- W3103545947 modified "2023-09-24" @default.
- W3103545947 title "Improved Lower Bounds for the 2-Page Crossing Numbers of $K_{m,n}$ and $K_n$ via Semidefinite Programming" @default.
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- W3103545947 doi "https://doi.org/10.1137/110852206" @default.
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