Boxicity and Cubicity of Asteroidal Triple free graphs [article]

Diptendu Bhowmick, L. Sunil Chandran
2009 arXiv   pre-print
An axis parallel d-dimensional box is the Cartesian product R_1 × R_2 × ... × R_d where each R_i is a closed interval on the real line. The boxicity of a graph G, denoted as (G), is the minimum integer d such that G can be represented as the intersection graph of a collection of d-dimensional boxes. An axis parallel unit cube in d-dimensional space or a d-cube is defined as the Cartesian product R_1 × R_2 × ... × R_d where each R_i is a closed interval on the real line of the form [a_i,a_i +
more » ... The cubicity of G, denoted as (G), is the minimum integer d such that G can be represented as the intersection graph of a collection of d-cubes. Let S(m) denote a star graph on m+1 nodes. We define claw number of a graph G as the largest positive integer k such that S(k) is an induced subgraph of G and denote it as . Let G be an AT-free graph with chromatic number χ(G) and claw number . In this paper we will show that (G) ≤χ(G) and this bound is tight. We also show that (G) ≤(G)(_2 +2) ≤ χ(G)(_2 +2). If G is an AT-free graph having girth at least 5 then (G) ≤ 2 and therefore (G) ≤ 2_2 +4.
arXiv:0812.0894v2 fatcat:mnrbwranvbayhl3eq652xbffqi