Chandran, Sunil L and Francis, Mathew and Sivadasan, Naveen
(2006)
*Geometric representations of graphs in low dimension.*
In: 12th Annual International Conference on Computing and Combinatorics (COCOON-2006), , August . 2006, Taiwan, Taipei.

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## Abstract

We give an efficient randomized algorithm to construct a box representation of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where $\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta + 2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We also show that our randomized algorithm can be derandomized to get a polynomial time deterministic algorithm. Though our general upper bound is in terms of maximum degree $\Delta$, we show that for almost all graphs on n vertices, its boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the average degree and c is a small constant. Also, we show that for any graph G, $\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b \sqrt{\ln n}$ for a constant b.

Item Type: | Conference Paper |
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Publisher: | Springer-Verlag Berlin |

Additional Information: | Copyright of this article belongs to Springer-Verlag Berlin. |

Keywords: | Boxicity;randomized algorithm;derandomization;random graph; intersection graphs |

Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |

Date Deposited: | 10 Nov 2011 05:54 |

Last Modified: | 10 Nov 2011 05:54 |

URI: | http://eprints.iisc.ac.in/id/eprint/41960 |

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