关于代数定义图类的速度
On the speed of algebraically defined graph classes
摘要：
一类图的速度是计算类内顶点集$\lbrace 1,\dots, n\rbrace$上的图的数量，作为$n$的函数。在本文中，我们调查这个功能的许多类的图，自然出现在离散几何，例如交图的部分或磁盘在平面上。虽然上界遵循沃伦的定理（一个变体的一个定理的米尔诺和汤姆），所有以前已知的下界是从特设建设非常具体的类。我们证明了一个一般定理，给出了一个本质上紧的下界的图的数量$\lbrace 1,\dots, n\rbrace$的边缘定义使用一个给定的有限列表的多项式的符号，假设这些多项式满足一些合理的条件。这特别意味着许多不同类别的交图的速度的下界，这基本上匹配已知的上界。我们的一般结果也给出了本质上紧下界计数包含秩序的各种家庭的几何对象，包括圆秩序和角度秩序。本文中提出的一些应用是新的，而其他恢复的结果Alon-Scheinerman，福克斯，McDiarmid-Müller和施。为了证明我们的结果，我们使用了一些工具，从代数几何和微分拓扑。
英文摘要：
The speed of a class of graphs counts the number of graphs on the vertex set $\lbrace 1,\dots, n\rbrace$ inside the class as a function of $n$. In this paper, we investigate this function for many classes of graphs that naturally arise in discrete geometry, for example intersection graphs of segments or disks in the plane. While upper bounds follow from Warren's theorem (a variant of a theorem of Milnor and Thom), all the previously known lower bounds were obtained from ad hoc constructions for very specific classes. We prove a general theorem giving an essentially tight lower bound for the number of graphs on $\lbrace 1,\dots, n\rbrace$ whose edges are defined using the signs of a given finite list of polynomials, assuming these polynomials satisfy some reasonable conditions. This in particular implies lower bounds for the speed of many different classes of intersection graphs, which essentially match the known upper bounds. Our general result also gives essentially tight lower bounds for counting containment orders of various families of geometric objects, including circle orders and angle orders. Some of the applications presented in this paper are new, whereas others recover results of Alon-Scheinerman, Fox, McDiarmid-Müller and Shi. For the proof of our result we use some tools from algebraic geometry and differential topology.
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