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- W1738343536 abstract "For a graph $G$, its emph{cubicity} $cub(G)$ is the minimum dimension $k$ such that $G$ is representable as the intersection graph of (axis--parallel) cubes in $k$--dimensional space. Chandran, Mannino and Oriolo showed that for a $d$--dimensional hypercube $H_d$, $frac{d-1}{log d} le cub(H_d) le 2d$. In this paper, we show that $cub(H_d) = Theta(frac{d}{log d})$.The parameter emph{boxicity} generalizes cubicity: the boxicity $box(G)$ of a graph $G$ is defined as the minimum dimension $k$ such that $G$ is representable as the intersection graph of axis parallel boxes in $k$ dimensional space. Since $box(G) le cub(G)$ for any graph $G$, our result implies that $box(H_d) = O(frac{d}{log d})$. The problem of determining a non-trivial lower bound for $box(H_d)$ is left open." @default.
- W1738343536 created "2016-06-24" @default.
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- W1738343536 date "2006-05-10" @default.
- W1738343536 modified "2023-09-27" @default.
- W1738343536 title "On the Boxicity and Cubicity of Hypercubes" @default.
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