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- W2123774049 abstract "The boxicity (respectively cubicity ) of a graph G is the least integer k such that G can be represented as an intersection graph of axis-parallel k -dimensional boxes (respectively k -dimensional unit cubes) and is denoted by box ( G ) (respectively cub ( G ) ). It was shown by Adiga and Chandran (2010) that for any graph G , cub ( G ) ≤ box ( G ) ⌈ log 2 α ( G ) ⌉ , where α ( G ) is the maximum size of an independent set in G . In this note we show that cub ( G ) ≤ 2 ⌈ log 2 χ ( G ) ⌉ box ( G ) + χ ( G ) ⌈ log 2 α ( G ) ⌉ , where χ ( G ) is the chromatic number of G. This result can provide a much better upper bound than that of Adiga and Chandran for graph classes with bounded chromatic number. For example, for bipartite graphs we obtain cub ( G ) ≤ 2 ( box ( G ) + ⌈ log 2 α ( G ) ⌉ ) . Moreover, we show that for every positive integer k , there exist graphs with chromatic number k such that for every ϵ > 0 , the value given by our upper bound is at most ( 1 + ϵ ) times their cubicity. Thus, our upper bound is almost tight." @default.
- W2123774049 created "2016-06-24" @default.
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- W2123774049 date "2016-02-01" @default.
- W2123774049 modified "2023-09-29" @default.
- W2123774049 title "Upper bound on cubicity in terms of boxicity for graphs of low chromatic number" @default.
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- W2123774049 doi "https://doi.org/10.1016/j.disc.2015.09.007" @default.
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