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arXiv 1801.04559math.CO
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多分支图类的渐近Enumeration

Asymptotic Enumeration of Graph Classes with Many Components

Konstantinos Panagiotou, Leon Ramzews

发表信息：
2018 Proceedings of the Fifteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), pp. 133--142
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摘要：

我们考虑图类$\mathcal G$，其中每个图都有组件的一个类$\mathcal{C}$的连接图。我们提供了一个框架的渐近研究$\lvert\mathcal{G}_{n,N}\rvert$，在$\mathcal{G}$与$n$顶点和$N:=\lfloorλn\rfloor$组件，其中$λ\in(0,1)$图的数量。假设在$\mathcal{C}$中具有$n$顶点的图的数目满足\begin{align*} \lvert \mathcal{C}_n\rvert\sim b n^{-(1+α)}ρ^{-n}n!, \quad n\to \infty \end{align*}对于一些$b,ρ>0$和$α>1$ --在图枚举中通常遇到的一个属性--我们证明了对于明确给定的$c(λ),f(λ),g(λ)$和$h(λ)$，\begin{align*} \lvert\mathcal{G}_{n,N}\rvert\sim c(λ) n^{f(λ)} (\log n)^{g(λ)} ρ^{-n}h(λ)^{N}\frac{n!}{N!}, \quad n\to \infty \end{align*}。这些功能是分段连续的不连续性在一个临界值$λ^{*}$，我们也确定。在我们的方法的中心思想是随机采样对象的$\cal G$所谓的玻尔兹曼发电机，以翻译枚举问题的分析iid随机变量。通过这一点，我们能够利用局部极限定理和大偏差结果众所周知的概率论来证明我们的主张。主要结果是制定一般的组合类满足SET-建设。

英文摘要：

We consider graph classes $\mathcal G$ in which every graph has components in a class $\mathcal{C}$ of connected graphs. We provide a framework for the asymptotic study of $\lvert\mathcal{G}_{n,N}\rvert$, the number of graphs in $\mathcal{G}$ with $n$ vertices and $N:=\lfloorλn\rfloor$ components, where $λ\in(0,1)$. Assuming that the number of graphs with $n$ vertices in $\mathcal{C}$ satisfies \begin{align*} \lvert \mathcal{C}_n\rvert\sim b n^{-(1+α)}ρ^{-n}n!, \quad n\to \infty \end{align*} for some $b,ρ>0$ and $α>1$ -- a property commonly encountered in graph enumeration -- we show that \begin{align*} \lvert\mathcal{G}_{n,N}\rvert\sim c(λ) n^{f(λ)} (\log n)^{g(λ)} ρ^{-n}h(λ)^{N}\frac{n!}{N!}, \quad n\to \infty \end{align*} for explicitly given $c(λ),f(λ),g(λ)$ and $h(λ)$. These functions are piecewise continuous with a discontinuity at a critical value $λ^{*}$, which we also determine. The central idea in our approach is to sample objects of $\cal G$ randomly by so-called Boltzmann generators in order to translate enumerative problems to the analysis of iid random variables. By that we are able to exploit local limit theorems and large deviation results well-known from probability theory to prove our claims. The main results are formulated for generic combinatorial classes satisfying the SET-construction.

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