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arXiv 1801.01047math.CO
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几乎所有图的最大诱导密度及其诱导数

On the exact maximum induced density of almost all graphs and their inducibility

Raphael Yuster

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27 pages
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摘要：

设$H$是$h$顶点上的图。图$G$中$H$的诱导拷贝数由$i_H(G)$表示。令$i_H(n)$表示在所有具有$n$顶点的图$G$上所取的$i_H(G)$的最大值。设$f(n,h) = Π_{i}^h a_i$，其中$\sum_{i=1}^h a_i = n$和$a_i$尽可能相等。令$g(n,h) = f(n,h) + \sum_{i=1}^h g(a_i,h)$。证明了对于几乎所有的$h$顶点上的图$H$，对所有的$n \le 2^{\sqrt{h}}$都成立$i_H(n)=g(n,h)$.更确切地说，我们定义了一个明确的图形属性${\cal P}_h$，当满足$H$，保证$i_H(n)=g(n,h)$的所有$n \le 2^{\sqrt{h}}$。特别地，证明了$h$顶点上的随机图以概率$1-o_h(1)$满足${\cal P}_h$。此外，所有的极值$n$ -顶点图产生$i_H(n)$在上述范围内被确定。我们还证明了一个稳定的结果。对于$H \in {\cal P}_h$和一个有$n \le 2^{\sqrt{h}}$顶点的图$G$满足$i_H(G) \ge f(n,h)$，则$G$必须是由$H$的平衡爆破通过在爆破部分内添加一些边而得到的. $H$的{ \em诱导}为$i_H = \lim_{n \rightarrow \infty} i_H(n)/\binom{n}{h}$。已知$i_H \ge h!/(h^h-h)$对所有图$H$和随机图$H$几乎必然满足$i_H \le h^{3\log h}h!/(h^h-h)$ .我们改进了这个上界几乎匹配的下界。证明了满足${\cal P}_h$的图$H$有$i_H =(1+O(h^{-h^{1/3}}))h!/(h^h-h)$ .

英文摘要：

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = Π_{i}^h a_i$ where $\sum_{i=1}^h a_i = n$ and the $a_i$ are as equal as possible. Let $g(n,h) = f(n,h) + \sum_{i=1}^h g(a_i,h)$. It is proved that for almost all graphs $H$ on $h$ vertices it holds that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. More precisely, we define an explicit graph property ${\cal P}_h$ which, when satisfied by $H$, guarantees that $i_H(n)=g(n,h)$ for all $n \le 2^{\sqrt{h}}$. It is proved, in particular, that a random graph on $h$ vertices satisfies ${\cal P}_h$ with probability $1-o_h(1)$. Furthermore, all extremal $n$-vertex graphs yielding $i_H(n)$ in the aforementioned range are determined. We also prove a stability result. For $H \in {\cal P}_h$ and a graph $G$ with $n \le 2^{\sqrt{h}}$ vertices satisfying $i_H(G) \ge f(n,h)$, it must be that $G$ is obtained from a balanced blowup of $H$ by adding some edges inside the blowup parts. The {\em inducibility} of $H$ is $i_H = \lim_{n \rightarrow \infty} i_H(n)/\binom{n}{h}$. It is known that $i_H \ge h!/(h^h-h)$ for all graphs $H$ and that a random graph $H$ satisfies almost surely that $i_H \le h^{3\log h}h!/(h^h-h)$. We improve upon this upper bound almost matching the lower bound. It is shown that a graph $H$ which satisfies ${\cal P}_h$ has $i_H =(1+O(h^{-h^{1/3}}))h!/(h^h-h)$.

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