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Bounds for the crossing number of the N-cube. (English) Zbl 0722.05028
Let \(Q_ n\) denote the n-dimensional hypercube. In this paper we derive upper and lower bounds for the crossing number \(\nu (Q_ n)\), i.e., the minimum number of edge-crossings in any planar drawing of \(Q_ n\). The upper bound is close to a result conjectured by Eggleton and Guy and the lower bound is a significant improvement over what was previously known. Let \(N=2^ n\) be the number of vertices of \(Q_ n\). We show that \(\nu (Q_ n)<\frac{1}{6}N^ 2\). For the lower bound we prove that \(\nu (Q_ n)=\Omega (N(lg N)^{c lg lg N})\), where \(c>0\) is a constant and lg is the logarithm base 2. The best lower bound using standard arguments is \(\nu (Q_ n)=\Omega (N(lg N)^ 2)\). The lower bound is obtained by constructing a large family of homeomorphs of a subcube with the property that no given pair of edges can appear in more than a constant number of the homeomorphs.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C35 Extremal problems in graph theory
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