A -box , where each is a closed interval on the real line, is defined to be the Cartesian product . If each is a unit-length interval, we call a -cube. The boxicity of a graph , denoted as , is the minimum integer such that is an intersection graph of -boxes. Similarly, the cubicity of , denoted as , is the minimum integer such that is an intersection graph of -cubes.
It was shown in [L. Sunil Chandran, Mathew C. Francis, Naveen Sivadasan. Cubicity and bandwidth. Graphs and Combinatorics 29 (1) (2013) 45–69] that, for a graph with vertices and maximum degree , . In this paper we show the following:
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For a -degenerate graph , . This bound is tight up to a constant factor.
Since
is at most
and can be much lower, this clearly is an asymptotically stronger result. Moreover, we have an efficient deterministic algorithm that runs in
time to output an
-dimensional cube representation for
. The above result has the following interesting consequences:
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If the crossing number of a graph is , then is . This bound is tight up to a factor of . We also show that if has vertices, then is .
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Let denote the poset dimension of a partially ordered set . We show that , where denotes the degeneracy of the underlying comparability graph of .
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We show that the cubicity of almost all graphs in the model is , where denotes the average degree of the graph under consideration.