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The cubicity of hypercube graphs

Chandran, L Sunil and Sivadasan, Naveen (2008) The cubicity of hypercube graphs. In: Discrete Mathematics, 308 (23). pp. 5795-5800.

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Abstract

For a graph G, its cubicity View the MathML source is the minimum dimension k such that Gis representable as the intersection graph of (axis-parallel) cubes in k-dimensional space. (A k-dimensional cube is a Cartesian product R1×R2×cdots, three dots, centered×Rk, where Ri is a closed interval of the form [ai,ai+1] on the real line.) Chandran et al. [L.S. Chandran, C. Mannino, G. Oriolo, On the cubicity of certain graphs, Information Processing Letters 94 (2005) 113–118] showed that for a d-dimensional hypercube Hd, View the MathML source. In this paper, we use the probabilistic method to show that View the MathML source. The parameter boxicity generalizes cubicity: the boxicity View the MathML source of a graph G is defined as the minimum dimension k such that G is representable as the intersection graph of axis-parallel boxes in k-dimensional space. Since View the MathML source for any graph G, our result implies that View the MathML source. The problem of determining a non-trivial lower bound for View the MathML source is left open.

Item Type: Journal Article
Additional Information: Copyright of this article belongs to Elsevier Science.
Keywords: Cubicity;Boxicity;Hypercube.
Department/Centre: Division of Electrical Sciences > Computer Science & Automation (Formerly, School of Automation)
Date Deposited: 21 Aug 2009 08:29
Last Modified: 19 Sep 2010 04:57
URI: http://eprints.iisc.ernet.in/id/eprint/17549

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