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- W2122913936 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 ." @default.
- W2122913936 created "2016-06-24" @default.
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- W2122913936 date "2010-06-01" @default.
- W2122913936 modified "2023-09-28" @default.
- W2122913936 title "Boxicity and cubicity of asteroidal triple free graphs" @default.
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- W2122913936 doi "https://doi.org/10.1016/j.disc.2010.01.020" @default.
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