# Layout Volumes of the Hypercube

Torok, Lubomir and Vrt'o, Imrich (2004) Layout Volumes of the Hypercube. In: Graph Drawing 12th International Symposium, GD 2004, September 29-October 2, 2004, New York, NY, USA , pp. 414-424 (Official URL: http://dx.doi.org/10.1007/978-3-540-31843-9_42).

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## Abstract

We study 3-dimensional layouts of the hypercube in a 1-active layer and a general model. The problem can be understood as a graph drawing problem in 3D space and was addressed at Graph Drawing 2003 [5]. For both models we prove general lower bounds which relate volumes of layouts to a graph parameter called cutwidth. Then we propose tight bounds on volumes of layouts of N-vertex hypercubes. Especially, we have ${\rm VOL}_{1-AL}(Q_{\log N})= \frac{2}{3}N^{\frac{3}{2}}\log N +O(N^{\frac{3}{2}}),$ for even log N and ${\rm VOL}(Q_{\log N})=\frac{2\sqrt{6}}{9}N^{\frac{3}{2}}+O(N^{4/3}\log N),$ for log N divisible by 3. The 1-active layer layout can be easily extended to a 2-active layer (bottom and top) layout which improves a result from [5]. This research was partially supported by the VEGA grant No. 2/3164/23.

Item Type: Conference Paper 10.1007/978-3-540-31843-9_42 A General Literature > A.001 Introductory and Survey 612

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