Layout volumes of the hypercube

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. 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 VOL1-AL(Qlog N) = 2/3 N3/2 log N + O(N3/2), for even log N and VOL(Qlog N) = 2√6/9 N 3/2+O(N4/3logN), 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]. © Springer-Verlag Berlin Heidelberg 2004.