发表机构
Nankai University; Royal Holloway, University of London（南开大学; 伦敦大学皇家霍洛威学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对无几何交叉的嵌套环族的最少边数问题，改进了此前的通用界，得到对固定$k\ge3$的更优上界，特别给出$n(\log n)^{o(1)}$的界。
AI 中文摘要
图中的环$C_1,\ldots,C_k$若两两边不交，满足$V(C_k)\subseteq\cdots\subseteq V(C_1)$，且每对连续环对内环顶点的循环顺序（不计反转）相同，则称为无几何交叉的嵌套环。设$f_k(n)$为使每个$n$顶点图都包含此类环族的最少边数。针对Erdős提出的双环问题，Gil Fernández、Kim、Kim和Liu证明$f_2(n)=O(n)$，并询问对每个固定$k$是否有$f_k(n)=O_k(n)$。Xu、Zeng和Zhang近期得到首个通用界：对每个固定$k\ge3$，$f_k(n)=O_k\bigl(n(\log n)^{k-1}(\log\log n)^{k-3}\bigr)$。本文证明，对每个固定$k\ge3$，$f_k(n)=O_k\left(n\frac{(\log\log n)^2}{\log\log\log n}\right)$，故特别地$f_k(n)\le n(\log n)^{o(1)}$，其中该依赖$n$的迭代对数因子对任意固定环数均为同一形式。
英文摘要
Cycles $C_1,\ldots,C_k$ in a graph are called nested without geometric crossings if they are pairwise edge-disjoint, $V(C_k)\subseteq\cdots\subseteq V(C_1)$, and each pair of consecutive cycles induces the same cyclic order on the vertices of the inner cycle, up to reversal. Let $f_k(n)$ be the least number of edges that forces such a family in every $n$-vertex graph. Gil Fernández, Kim, Kim and Liu proved that $f_2(n)=O(n)$, answering a question of Erdős, and asked whether $f_k(n)=O_k(n)$ for every fixed $k$. We prove this for all $k$. The proof selects the inner cycles together with a disjoint subgraph that supplies their external neighbours. A reselection argument gives disjoint paths from every inner-cycle vertex to any sufficiently large target set. Sublinear expansion and a rooted clique minor then allow the vertices to be joined in the required cyclic order.
