Chandran, LS and Mathew, KA (2009) An upper bound for Cubicity in terms of Boxicity. In: Discrete Mathematics, 309 (8). pp. 25712574.
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Abstract
An axisparallel bdimensional box is a Cartesian product R1 x R2 x ... x Rb where each Ri (for 1 <= i <= b) is a closed interval of the form [a(i), b(i)] on the real line. The boxicity of any graph G, box(G) is the minimum positive integer b such that G can be represented as the intersection graph of axisparallel bdimensional boxes. A bdimensional cube is a Cartesian product R1 x R2 x ... x Rb, where each Ri (for 1 <= i <= b) is a closed interval of the form [a(i), a(i) + 1] on the real line. When the boxes are restricted to be axisparallel cubes in bdimension, the minimum dimension b required to represent the graph is called the cubicity of the graph (denoted by cub(G)). In this paper we prove that cub(G) <= inverted right perpendicularlog(2) ninverted left perpendicular box(G), where n is the number of vertices in the graph. We lso show that this upper bound is tight.Some immediate consequences of the above result are listed below:1. Planar graphs have cubicity at most 3inverted right rpendicularlog(2) ninvereted left perpendicular.2. Outer planar graphs have cubicity at most 2inverted right perpendicularlog(2) ninverted left perpendicular.3. Any graph of treewidth tw has cubicity at most (tw + 2) inverted right perpendicularlog(2) ninverted left erpendicular. Thus, chordal graphs have cubicity at most (omega + 1) inverted right perpendicularlog(2) ninverted left perpendicular and circular arc graphs have cubicity at most (2 omega + 1)inverted right perpendicularlog(2) ninverted left perpendicular, where omega is the clique number.
Item Type:  Journal Article 

Publication:  Discrete Mathematics 
Publisher:  Elsevier Science 
Additional Information:  Copyright for this article belongs to Elsevier Science. 
Keywords:  Cubicity;Boxicity;Interval graph;Indifference graph 
Department/Centre:  Division of Electrical Sciences > Computer Science & Automation 
Date Deposited:  28 Aug 2018 15:29 
Last Modified:  28 Aug 2018 15:29 
URI:  http://eprints.iisc.ac.in/id/eprint/60563 
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An upper bound for Cubicity in terms of Boxicity. (deposited 14 Dec 2009 08:00)
 An upper bound for Cubicity in terms of Boxicity. (deposited 30 Aug 2018 15:41)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:33)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:33)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:33)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:30)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:29)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:29)
 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:29)
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 An upper bound for Cubicity in terms of Boxicity. (deposited 28 Aug 2018 15:29)
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