发表机构
Institute for Basic Science (IBS); Extremal Combinatorics and Probability Group (ECOPRO)（基础科学研究院; 极值组合与概率研究组）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造度为(1/2+o(1))log²n的正则次线性扩张器，否定了Montgomery关于足够大度正则次线性扩张器必为哈密顿图的猜想，揭示了log²n是阻碍环覆盖的自然度尺度。
AI 中文摘要
次线性扩张足够弱，可从任意图中提取且保留几乎所有平均度，但已被证明足够强，能在许多稀疏极值问题中强制形成全局结构。Letzter、Methuku和Sudakov[JLMS 2026]开发了方法，可在足够稠密的正则次线性扩张器中生成近似哈密顿环；随后Montgomery[ICM 2026]猜想，每个足够大（但为常数）度的d-正则次线性扩张器都是哈密顿图。我们以强形式否定该猜想，构造了n顶点、度为d=(1/2+o(1))log²n的d-正则次线性扩张器，其甚至不存在覆盖正比例顶点的环。该构造将双正则拉马努金图的一侧膨胀为几乎完全块，同时保持另一侧为独立集；拉马努金关联图为部分块与分隔点的任意组合提供扩张性证明，而独立侧形成稀疏顶点分隔器，阻止环访问足够多的块。该构造也解释了为何log²n是此阻碍的自然度尺度。
英文摘要
Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] proved the existence of nearly Hamilton cycles in sufficiently dense regular sublinear expanders. Montgomery [ICM 2026] subsequently conjectured that, every $d$-regular sublinear expander with $d$ sufficiently large (but constant) is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with $d=\left(\frac12+o(1)\right)\log^2 n$, which can forbid any cycle covering an arbitrarily small given positive constant portion of its vertices. The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks and keeps the other side as a sparse vertex separator. We also prove a similar statement for a closely related notion of edge expanders. For every sufficiently small $γ>0$, there is an infinite family of $n$-vertex $d$-regular $γ$-edge-expanders with $d=Θ(γ^{-1})$ and circumference $O(γn)$, matching the standard lower bound $Ω(γn)$. The construction also comes from an expanding regular core such that each vertex has an almost-complete graph attached to it. The core guarantees edge expansion, while the single-vertex attachments confine every cycle.
Comments14pages, 4 figures
