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 |
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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 |

Depositing User: | Id for Latest eprints |

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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