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# Boxicity and cubicity of asteroidal triple free graphs

Bhowmick, Diptendu and Chandran, Sunil L (2010) Boxicity and cubicity of asteroidal triple free graphs. In: Discrete Mathematics, 310 (10-11). pp. 1536-1543. PDF boxicity.pdf - Published Version Restricted to Registered users only Download (339kB) | Request a copy
Official URL: http://dx.doi.org/10.1016/j.disc.2010.01.020

## Abstract

The boxicity of a graph G, denoted box(G), is the least integer d such that G is the intersection graph of a family of d-dimensional (axis-parallel) boxes. The cubicity, denoted cub(G), is the least dsuch that G is the intersection graph of a family of d-dimensional unit cubes. An independent set of three vertices is an asteroidal triple if any two are joined by a path avoiding the neighbourhood of the third. A graph is asteroidal triple free (AT-free) if it has no asteroidal triple. The claw number psi(G) is the number of edges in the largest star that is an induced subgraph of G. For an AT-free graph G with chromatic number chi(G) and claw number psi(G), we show that box(G) &lt;= chi(C) and that this bound is sharp. We also show that cub(G) &lt;= box(G)([log(2) psi(G)] + 2) &lt;= chi(G)([log(2) psi(G)] + 2). If G is an AT-free graph having girth at least 5, then box(G) &lt;= 2, and therefore cub(G) &lt;= 2 [log(2) psi(G)] + 4. (c) 2010 Elsevier B.V. All rights reserved.

Item Type: Journal Article Copyright of this article belongs to Elsevier Science. Boxicity; Cubicity; Chordal dimension; Asteroidal triple free graph; Chromatic number; Claw number Division of Electrical Sciences > Computer Science & Automation Ms G Yashodha 04 Jun 2010 04:33 19 Sep 2010 06:06 http://eprints.iisc.ac.in/id/eprint/27719

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