Boxicity and cubicity of asteroidal triple free graphs

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 d such 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 ψ ( 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 χ ( G ) and claw number ψ ( G ) , we show that box ( G ) ≤ χ ( G ) and that this bound is sharp. 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 .