arXiv 2007.022162020-07math.CO

指定度数的随机图的子图概率及其在色数和连通性中的应用

Subgraph probability of random graphs with specified degrees and applications to chromatic number and connectivity

Pu Gao, Yuval Ohapkin

AI总结：

本文针对指定度数的随机图，给出任意边集联合概率的上下界，从而将配置模型分析直接转化，并应用于改进色数结果及刻画连通性相变。

摘要：

给定一个图形度序列 ${\bf d}=(d_1,\ldots, d_n)$，设 $G(n, {\bf d})$ 表示在顶点集 $[n]$ 上的均匀随机图，其中每个 $1\le i\le n$ 的顶点 $i$ 的度数为 $d_i$。我们给出了 $G(n,{\bf d})$ 中任意边集的联合概率的上界和下界。这些上界和下界近似于在配置模型中得到的结果，因此配置模型中的分析可以直接转化为 $G(n,{\bf d})$，而无需以配置模型产生简单图作为条件。文献中许多关于 $G(n,{\bf d})$ 的现有结果可以通过应用这一新的概率工具，以更简单的证明得到显著改进。我们给出的一个例子是关于 $G(n,{\bf d})$ 的色数。在另一个应用中，我们使用这些联合概率来研究 $G(n,{\bf d})$ 的连通性。当 $\Delta^2=o(M)$ 时，其中 $\Delta$ 是 ${\bf d}$ 的最大分量，我们完全刻画了 $G(n,{\bf d})$ 的连通性相变。当 $\Delta$ 不受限制时，我们也给出了 $G(n,{\bf d})$ 连通的充分条件。

英文摘要：

Given a graphical degree sequence ${\bf d}=(d_1,\ldots, d_n)$, let $G(n, {\bf d})$ denote a uniformly random graph on vertex set $[n]$ where vertex $ i$ has degree $d_i$ for every $1\le i\le n$. We give upper and lower bounds on the joint probability of an arbitrary set of edges in $G(n,{\bf d})$. These upper and lower bounds are approximately what one would get in the configuration model, and thus the analysis in the configuration model can be translated directly to $G(n,{\bf d})$, without conditioning on that the configuration model produces a simple graph. Many existing results of $G(n,{\bf d})$ in the literature can be significantly improved with simpler proofs, by applying this new probabilistic tool. One example we give is about the chromatic number of $G(n,{\bf d})$. In another application, we use these joint probabilities to study the connectivity of $G(n,{\bf d})$. When $Δ^2=o(M)$ where $Δ$ is the maximum component of ${\bf d}$, we fully characterise the connectivity phase transition of $G(n,{\bf d})$. We also give sufficient conditions for $G(n,{\bf d})$ being connected when $Δ$ is unrestricted.

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摘要：

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