论曲线图有限子图的复杂性
On the complexity of finite subgraphs of the curve graph
AI总结：
本文引入“嵌套复杂度长度”作为曲面曲线图诱导子图性质的新障碍，证明大半图和大型多部子图不满足该性质，并据此计算曲线图上密度。
摘要：
如果一个图是亏格为 g 且有 p 个穿孔的曲面的曲线图的诱导子图，我们就称该图具有性质 $\mathcal{P}_{g,p}$。两个著名的图不变量——色数和团数——可以为 $\mathcal{P}_{g,p}$ 提供障碍。我们引入了一个新的图不变量“嵌套复杂度长度”，它为 $\mathcal{P}_{g,p}$ 提供了一种新的障碍。对于曲线图而言，该不变量以图论语言刻画了曲面的拓扑复杂度；事实上，我们证明其值为 $6g-6+2p$，即为该曲面上极大多重曲线大小的两倍。作为推论，我们证明了大半图不具有 $\mathcal{P}_{g,p}$，并定量地推出：几乎所有通过色数和团数检验的有限图都不具有 $\mathcal{P}_{g,p}$。我们还分别从曲线图的一阶理论以及映射类群的 RAAG 子群（沿用 Kim 和 Koberda 的工作）角度重新解释了这一障碍。最后，我们证明了大型多部子图不可能具有 $\mathcal{P}_{g,p}$。这使我们能够计算曲线图的上密度，并得出结论：团大小、色数和嵌套复杂度长度并不足以确定 $\mathcal{P}_{g,p}$。
英文摘要：
We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the 'nested complexity length', which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large half-graphs do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large multipartite subgraphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.
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