随机正则图中的最大割与最大二分
Maximum cut and maximum bisection in random regular graphs
中文总结 AI 辅助
该研究证明在随机正则图中，最大割与最大二分期望值之差为亚线性，即限制割两侧等大小几乎不影响最优割，并利用配置模型和Huang定理得出二者极限密度相同。
中文摘要 AI 辅助
我们证明，对于每个固定的度数 $d$ 和偶数个顶点 $n>d$ 上的均匀随机简单 $d$-正则图 $G_{n,d}$，有 $$\u0415\u041caxCut(G_{n,d})-\u0415\u041caxBis(G_{n,d})=o(n).$$ 换言之，要求割的两侧具有完全相同的大小，仅使期望最优割改变亚线性数量的边。我们首先对配置模型证明这一点，通过比较 $n$ 顶点图上每种可能基数的割与相关 $2n$ 顶点图上的二分。一个重要的技术工具是 Huang 的插值定理（Huang, 2018）；为应用它，我们建立了当添加单条边时最优割变化的结构性质。该比较，连同浓度估计和 Huang 关于最大二分的收敛定理，表明最大割和最大二分具有相同的极限密度。对配置模型进行简单条件化，则得到均匀随机简单正则图的结果。
英文摘要
We prove that, for every fixed degree $d$ and a uniformly random simple $d$-regular graph $G_{n,d}$ on an even number $n>d$ of vertices, $$\mathbb{E}\operatorname{MaxCut}(G_{n,d})-\mathbb{E}\operatorname{MaxBis}(G_{n,d})=o(n).$$ In other words, requiring the two sides of a cut to have exactly the same size changes the expected optimal cut by only a sublinear number of edges. We prove this first for the configuration model by comparing cuts of each possible cardinality on an $n$-vertex graph with bisections of a related graph on $2n$ vertices. An important technical tool is Huang's interpolation theorem (Huang, 2018); to apply it, we establish a structural property of the change in the optimal cut when a single edge is added. This comparison, together with concentration estimates and Huang's convergence theorem for maximum bisection, shows that maximum cut and maximum bisection have the same limiting density. Conditioning the configuration model on being simple then gives the result for uniformly random simple regular graphs.
