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.
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Torok, L., Vrt’o, I. (2005). Layout Volumes of the Hypercube. In: Pach, J. (eds) Graph Drawing. GD 2004. Lecture Notes in Computer Science, vol 3383. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31843-9_42
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