# On the cubicity of bipartite graphs

Chandran, Sunil L and Das, Anita and Sivadasan, Naveen (2009) On the cubicity of bipartite graphs. In: Information Processing Letters, 109 (9). pp. 432-435.

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## Abstract

A unit cube in k-dimension (or a k-cube) is defined as the Cartesian product R-1 x R-2 x ... x R-k, where each R-i is a closed interval on the real line of the form [a(j), a(i), + 1]. The cubicity of G, denoted as cub(G), is the minimum k such that G is the intersection graph of a collection of k-cubes. Many NP-complete graph problems can be solved efficiently or have good approximation ratios in graphs of low cubicity. In most of these cases the first step is to get a low dimensional cube representation of the given graph. It is known that for graph G, cub(G) &lt;= left perpendicular2n/3right perpendicular. Recently it has been shown that for a graph G, cub(G) &gt;= 4(Delta + 1) In n, where n and Delta are the number of vertices and maximum degree of G, respectively. In this paper, we show that for a bipartite graph G = (A boolean OR B, E) with |A| = n(1), |B| = n2, n(1) &lt;= n(2), and Delta' = min {Delta(A),Delta(B)}, where Delta(A) = max(a is an element of A)d(a) and Delta(B) = max(b is an element of B) d(b), d(a) and d(b) being the degree of a and b in G, respectively , cub(G) &lt;= 2(Delta' + 2) bar left rightln n(2)bar left arrow. We also give an efficient randomized algorithm to construct the cube representation of G in 3 (Delta' + 2) bar right arrowIn n(2)bar left arrow dimension. The reader may note that in general Delta' can be much smaller than Delta.

Item Type: Journal Article Information Processing Letters Elsevier Science Copyright of this article belongs to Elsevier Science. Cubicity;Algorithms;Intersection graphs Division of Electrical Sciences > Computer Science & Automation 14 Dec 2009 07:06 28 Feb 2019 08:39 http://eprints.iisc.ac.in/id/eprint/19890