随机图的极值子图：一个推广
Extremal Subgraphs of Random Graphs: an Extended Version
摘要：
证明了存在一个常数$c >0$，使得只要$p \ge n^{-c}$，当$n$趋于无穷大时概率趋于1，随机图$G_{n,p}$的每个最大无三角形子图都是二部图.这回答了Babai，Simonovits和Spencer（Journal of图论，1990）的一个问题。证明是基于一个工具的独立利益：例如，我们表明，几乎所有的图与$M$边缘，其中$M >> n$的最大切割，是“几乎唯一的”。更准确地说，给定$G_{n,M}$的最大割$C$，我们可以通过在$C$的部分之间移动至多$O(\sqrt{n^3/M})$顶点来获得所有最大割。
英文摘要：
We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question of Babai, Simonovits and Spencer (Journal of Graph Theory, 1990). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with $M$ edges, where $M >> n$, is ``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M}$, we can obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices between the parts of $C$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
