偏序集维数和盒性的较好界
Better bounds for poset dimension and boxicity
摘要：
证明了相似图具有最大度$Δ$的偏序集的维数至多为$Δ\log^{1+o(1)} Δ$。这个结果改进了Füredi和Kahn的30年前的界，并且在最优的$\log^{o(1)}Δ$因子内。我们通过盒性的概念证明了这个结果。图$G$的“盒度”是最小整数$d$，使得$G$是$d$维轴对齐盒的交图。我们证明了具有最大度$Δ$的图的盒度至多为$Δ\log^{1+o(1)} Δ$，并且也在最优的$\log^{o(1)}Δ$因子之内.我们还证明了具有Euler亏格$g$的图的最大盒度是$Θ(\sqrt{g \log g})$，它解决了Esperet和Jaret的一个公开问题，并且紧到一个$O(1)$因子.
英文摘要：
We prove that the dimension of every poset whose comparability graph has maximum degree $Δ$ is at most $Δ\log^{1+o(1)} Δ$. This result improves on a 30-year old bound of Füredi and Kahn, and is within a $\log^{o(1)}Δ$ factor of optimal. We prove this result via the notion of boxicity. The "boxicity" of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $Δ$ has boxicity at most $Δ\log^{1+o(1)} Δ$, which is also within a $\log^{o(1)}Δ$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $Θ(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor.
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