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Upper bound on cubicity in terms of boxicity for graphs of low chromatic number
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For bipartite graphs with bounded chromatic number, this article showed that for every positive integer k > 0, the upper bound of the cubicity is at most Ω(1+ √ ϵ) times their cubicity, where ϵ is the cardinality of the maximum independent set in the graph.Abstract:
The boxicity (respectively cubicity) of a graph $G$ is the minimum non-negative 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 (Journal of Graph Theory, 65(4), 2010) that for any graph $G$, $cub(G) \le$ box$(G) \left \lceil \log_2 \alpha \right \rceil$, where $\alpha = \alpha(G)$ is the cardinality of the maximum independent set in $G$. In this note we show that $cub(G) \le 2 \left \lceil \log_2 \chi(G) \right \rceil box(G) + \chi(G) \left \lceil \log_2 \alpha(G) \right \rceil $. In general, 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 get, $cub(G) \le 2 (box(G) + \left \lceil \log_2 \alpha(G) \right \rceil )$.
Moreover we show that for every positive integer $k$, there exist graphs with chromatic number $k$, such that for every $\epsilon > 0$, the value given by our upper bound is at most $(1+\epsilon)$ times their cubicity. Thus, our upper bound is almost tight.read more
Citations
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References
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TL;DR: It is proved that all planar bipartite graphs have a grid representation, and some infinite families of graphs with no grid representation are exhibited-among them the point line incidence graph of projective planes.
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On the dimensions of ordered sets of bounded degree
TL;DR: Rodl and Trotter as discussed by the authors proved that there exists a constant c (c < 50) such that dim P < ck(log k)2. This result is nonconstructive, depending in part on Lovasz' local lemma.
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Contact representations of planar graphs with cubes
Stefan Felsner,Mathew C. Francis +1 more
TL;DR: It is proved that every planar graph has a representation using axis-parallel cubes in three dimensions in such a way that there is a cube corresponding to each vertex of the planargraph and two cubes have a non-empty intersection if and only if their corresponding vertices are adjacent.