Layout Volumes of the Hypercube

  • Lubomir Torok
  • Imrich Vrt’o
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)


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].


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Lubomir Torok
    • 1
  • Imrich Vrt’o
    • 2
  1. 1.Institute of Mathematics and Computer ScienceBanská BystricaSlovak Republic
  2. 2.Institute of MathematicsSlovak Academy of SciencesBratislavaSlovak Republic

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