AI 中文总结
该论文研究无三角形图中随机贪心独立集顶点数的方差，证明其上界为 e((n-2)/n)^2，并指出星图在树中唯一达到最大值，通过改进的估计和恒等式获得尖锐结果。
AI 中文摘要
以均匀随机顺序检查有限简单图的顶点，若某顶点的邻居均未被先前接受，则接受该顶点。设 $X_G$ 为被接受的顶点数。对于任意具有 $n\ge2$ 个顶点和 $e$ 条边的无三角形图，我们证明 $\operatorname{Var}(X_G)\le e((n-2)/n)^2$，且等号成立当且仅当图为无边图或连通星图。特别地，在给定阶数的树中，星图唯一地最大化方差，其值为 $(n-1)(n-2)^2/n^2$。已知的期望顶点删除稳定性已给出基本界 $\operatorname{Var}(X_G)\le e$。我们通过结合更强的中心化首选估计与全方差定律中的无三角形边数恒等式，获得了尖锐的有限阶改进。
英文摘要
Inspect the vertices of a finite simple graph in uniformly random order, accepting each vertex if none of its neighbors has previously been accepted. Let $X_G$ be the number of accepted vertices. For every triangle-free graph with $n\ge2$ vertices and $e$ edges, we prove $\operatorname{Var}(X_G)\le e((n-2)/n)^2$, with equality precisely for edgeless graphs and connected stars. In particular, among trees of a given order, the star uniquely maximizes the variance, with value $(n-1)(n-2)^2/n^2$. Known expected vertex-deletion stability already yields the elementary baseline $\operatorname{Var}(X_G)\le e$. We obtain the sharp finite-order refinement by combining a stronger centered first-choice estimate with a triangle-free edge-count identity in the law of total variance.
