盒性与拓扑不变量
Boxicity and topological invariants
摘要：
图$G=(V,E)$的盒度是存在$k$区间图$G_i=(V,E_i)$，$1 \le i \le k$，使得$E=E_1 \cap \cdots \cap E_k$的最小整数$k$.在本文的第一部分中，我们证明了$m$边上的每个图都有盒性$O(\sqrt{m \log m})$，这是渐近最佳可能的。我们利用这个结果来研究图的盒性和它们的Colin de Verdière不变量之间的联系，它们有许多相似之处。关于这两个参数的已知结果表明，对于任何图$G$，$G$的盒性至多是$G$的Colin de Verdière不变量，记为$μ(G)$。我们观察到每个图$G$都有盒度$O(μ(G)^4(\log μ(G))^2)$，同时也有图$G$有盒度$Ω(μ(G)\sqrt{\log μ(G)})$ .在本文的第二部分中，我们主要研究了可嵌入到Euler亏格$g$曲面上的图。我们证明了这些图具有盒性$O(\sqrt{g}\log g)$，而其中一些图具有盒性$Ω(\sqrt{g \log g})$。这改进了以前最好的已知的上界和下界。这些结果直接暗示了曲面图的邻接偏序集维数的一个近似最优界。
英文摘要：
The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap \cdots \cap E_k$. In the first part of this note, we prove that every graph on $m$ edges has boxicity $O(\sqrt{m \log m})$, which is asymptotically best possible. We use this result to study the connection between the boxicity of graphs and their Colin de Verdière invariant, which share many similarities. Known results concerning the two parameters suggest that for any graph $G$, the boxicity of $G$ is at most the Colin de Verdière invariant of $G$, denoted by $μ(G)$. We observe that every graph $G$ has boxicity $O(μ(G)^4(\log μ(G))^2)$, while there are graphs $G$ with boxicity $Ω(μ(G)\sqrt{\log μ(G)})$. In the second part of this note, we focus on graphs embeddable on a surface of Euler genus $g$. We prove that these graphs have boxicity $O(\sqrt{g}\log g)$, while some of these graphs have boxicity $Ω(\sqrt{g \log g})$. This improves the previously best known upper and lower bounds. These results directly imply a nearly optimal bound on the dimension of the adjacency poset of graphs on surfaces.
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