AI 中文总结
该研究构造了度远高于log²n尺度的无穷多非哈密顿d-正则(ε,d)-扩张器，否定了Chen等人提出的关于正则次线性扩张器哈密顿性的问题。
AI 中文摘要
Letzter、Methuku和Sudakov证明了足够正则的次线性扩张器包含几乎生成的环和路径。随后Montgomery猜想，每个足够大度的正则次线性扩张器都是哈密顿的。最近，Chen、Liu、Wei和Yang通过构造阶为n、度为d=(1/2+o(1))log²n且围长小的d-正则次线性扩张器，否定了该猜想。他们进一步提出问题：对每个固定ε>0，是否存在常数C=C(ε)，使得每个足够大的n顶点d-正则(ε,d)-扩张器（满足d≥Clog²n）都是哈密顿的。我们通过构造，对每个足够小的固定ε>0，存在无穷多非哈密顿的d-正则(ε,d)-扩张器，其度远高于log²n尺度，从而否定了该问题。
英文摘要
Letzter, Methuku and Sudakov proved that sufficiently regular sublinear expanders contain nearly spanning cycles and paths. Montgomery subsequently conjectured that every regular sublinear expander of sufficiently large degree is hamiltonian. Recently, Chen, Liu, Wei and Yang disproved this conjecture by constructing $d$-regular sublinear expanders of order $n$ with $d=(1/2+o(1))\log^2 n$ and small circumference. They further posed the problem of determining whether, for every fixed $\varepsilon>0$, there exists a constant $C=C(\varepsilon)$ such that every sufficiently large $n$-vertex $d$-regular $(\varepsilon,d)$-expander with $d\ge C\log^2 n$ is hamiltonian. We answer this problem negatively by constructing, for every sufficiently small fixed $\varepsilon>0$, infinitely many nonhamiltonian $d$-regular $(\varepsilon,d)$-expanders with degrees far above the $\log^2 n$ scale.
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