Layout Volumes of the Hypercube

We study 3-dimensional layouts of the hypercube in a 1-active layer and 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 cutwidth. Then we propose tight bounds on volumes of layouts of N-vertex hypercubes. Especially, we have VOL1−AL(Qlog N ) = 2 3 N 3 2 log N + O(N 3 2 ), for even log N and VOL(Qlog N ) = 2 √ 6 9 N 3 2 + O(N4/3 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].

• Type: Conference
• Year: 2004
• Where: GD
• Authors: Lubomir Torok, Imrich Vrto
• Keywords: 1-active Layer, GD 2004, Graph Drawing, Graph Drawing Problem