Boxicity and cubicity of asteroidal triple free graphs

An axis parallel d-dimensional box is the Cartesian product R1×R2×· · ·×Rd where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(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 R1 ×R2 × · · · ×Rd where each Ri is a closed interval on the real line of the form [ai, ai+1]. The 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. An independent set of three vertices is called an asteroidal triple if between each pair in the triple there exists a path which avoids the neighbourhood of the third. A graph is said to be Asteroidal Triple free (AT-free for short) if it does not contain an asteroidal triple. The class of AT-free graphs is a reasonably large one, which properly contains the class of interval graphs, trapezoid graphs, permutation graphs, cocomparability graphs etc. 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 ψ(G). Let G be an AT-free graph with chromatic number χ(G) and claw number ψ(G). In this paper we will show that box(G) ≤ χ(G) and this bound is tight. We also show that cub(G) ≤ box(G)(⌈log 2 ψ(G)⌉ + 2) ≤ χ(G)(⌈log 2 ψ(G)⌉ + 2). If G is an AT-free graph having girth at least 5 then box(G) ≤ 2 and therefore cub(G) ≤ 2 ⌈log 2 ψ(G)⌉+ 4.