An axis parallel d-dimensional box is the Cartesian product R1βΓR2βΓ...ΓRdβ where each Riβ is a closed interval on the real line.
The {\it boxicity} of a graph G, denoted as \boxi(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 R1βΓR2βΓ...ΓRdβ where each Riβ is a closed interval on the
real line of the form [aiβ,aiβ+1]. The {\it cubicity} of G, denoted as
\cub(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 {\it 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 \claw.
Let G be an AT-free graph with chromatic number Ο(G) and claw number
\claw. In this paper we will show that \boxi(G) \leq \chi(G) and this bound
is tight. We also show that \cub(G) \leq \boxi(G)(\ceil{\log_2 \claw} +2)β€\chi(G)(\ceil{\log_2 \claw} +2). If G is an AT-free graph having
girth at least 5 then \boxi(G) \leq 2 and therefore \cub(G) \leq
2\ceil{\log_2 \claw} +4.Comment: 15 pages: We are replacing our earlier paper regarding boxicity of
permutation graphs with a superior result. Here we consider the boxicity of
AT-free graphs, which is a super class of permutation graph