Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

Chandran, Sunil L and Mathew, Rogers and Rajendraprasad, Deepak (2016) Upper bound on cubicity in terms of boxicity for graphs of low chromatic number. In: DISCRETE MATHEMATICS, 339 (2). pp. 443-446.

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Abstract

The boxicity (respectively cubicity) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k-dimensional boxes (respectively k-dimensional unit cubes) and is denoted by box(G) (respectively cub(G)). It was shown by Adiga and Chandran (2010) that for any graph G, cub(G) <= box(G) log(2) alpha(G], where alpha(G) is the maximum size of an independent set in G. In this note we show that cub(G) <= 2 log(2) X (G)] box(G) + X (G) log(2) alpha(G)], where x (G) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub(G) <= 2(box(G) + log(2) alpha(G)] Moreover, we show that for every positive integer k, there exist graphs with chromatic number k such that for every epsilon > 0, the value given by our upper bound is at most (1 + epsilon) times their cubicity. Thus, our upper bound is almost tight. (c) 2015 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Publication:	DISCRETE MATHEMATICS
Publisher:	ELSEVIER SCIENCE BV
Additional Information:	Copy right for this article belongs to the ELSEVIER SCIENCE BV, PO BOX 211, 1000 AE AMSTERDAM, NETHERLANDS
Keywords:	Boxicity; Cubicity; Chromatic number
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	10 Feb 2016 05:43
Last Modified:	10 Feb 2016 05:43
URI:	http://eprints.iisc.ac.in/id/eprint/53191

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