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# LOWER BOUNDS FOR BOXICITY

Adiga, Abhijin and Chandran, Sunil and Sivadasan, NaveenL (2014) LOWER BOUNDS FOR BOXICITY. In: COMBINATORICA, 34 (6). pp. 631-655. PDF com_34-6_631_2014.pdf - Published Version Restricted to Registered users only Download (567kB) | Request a copy
Official URL: http://dx.doi.org/ 10.1007/s00493-011-2981-0

## Abstract

An axis-parallel b-dimensional box is a Cartesian product R-1 x R-2 x ... x R-b where R-i is a closed interval of the form a(i),b(i)] on the real line. For a graph G, its boxicity box(G) is the minimum dimension b, such that G is representable as the intersection graph of boxes in b-dimensional space. Although boxicity was introduced in 1969 and studied extensively, there are no significant results on lower bounds for boxicity. In this paper, we develop two general methods for deriving lower bounds. Applying these methods we give several results, some of which are listed below: 1. The boxicity of a graph on n vertices with no universal vertices and minimum degree delta is at least n/2(n-delta-1). 2. Consider the g(n,p) model of random graphs. Let p <= 1 - 40logn/n(2.) Then with high `` probability, box(G) = Omega(np(1 - p)). On setting p = 1/2 we immediately infer that almost all graphs have boxicity Omega(n). Another consequence of this result is as follows: For any positive constant c < 1, almost all graphs on n vertices and m <= c((n)(2)) edges have boxicity Omega(m/n). 3. Let G be a connected k-regular graph on n vertices. Let lambda be the second largest eigenvalue in absolute value of the adjacency matrix of G. Then, the boxicity of G is a least (kappa(2)/lambda(2)/log(1+kappa(2)/lambda(2))) (n-kappa-1/2n). 4. For any positive constant c 1, almost all balanced bipartite graphs on 2n vertices and m <= cn(2) edges have boxicity Omega(m/n).

Item Type: Journal Article COMBINATORICA SPRINGER HEIDELBERG Copy right for this article belongs to the SPRINGER HEIDELBERG, TIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY Division of Electrical Sciences > Computer Science & Automation 24 Feb 2015 06:41 24 Feb 2015 06:41 http://eprints.iisc.ac.in/id/eprint/50963 View Item