发表机构
ETH Zürich（苏黎世联邦理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出稀疏化-喷洒技术，证明坚韧图等图类的随机子图保持鲁棒性，并据此构造几乎哈密顿圈，解决Chvátal和Lovász的长期猜想。
AI 中文摘要
我们发展了稀疏化-喷洒技术，这是一种在随机顶点采样下证明图性质鲁棒性的通用方法。利用该方法，我们证明了坚韧图、高度连通的顶点传递图以及近正则次线性扩展图的随机诱导子图以非常高的概率保持强连通性或扩展性质。我们还证明了每个$k$-连通图，当$k=\omega(\log n)$时，包含一个$\Omega(k)$-连通的生成二分图。利用这些鲁棒性结果，我们进一步开发了一个通用框架，用于从随机采样的高度连通子图构造几乎哈密顿圈。作为结果，我们证明了当坚韧度或度数呈多对数级大时，坚韧图、连通顶点传递图和近正则扩展图包含长度至少为$(1-o(1))n$的圈。这给出了Chvátal和Lovász关于坚韧图和顶点传递图哈密顿性的长期猜想的新近解。
英文摘要
We develop the thinning--sprinkling technique, a general method for proving robustness of graph properties under random vertex sampling. Using it, we show that random induced subgraphs of tough graphs, high-degree connected vertex-transitive graphs, and nearly regular sublinear expanders retain strong connectivity or expansion properties with very high probability. We also prove that every $k$-connected graph with $k=ω(\log n)$ contains a spanning bipartite subgraph that is $Ω(k)$-connected. Using these robustness results, we further develop a general framework for constructing almost Hamilton cycles from randomly sampled highly connected subgraphs. As a consequence, we show that tough graphs, connected vertex-transitive graphs and nearly regular expanders contain a cycle of length at least $(1-o(1))n$ whenever the toughness or degree is polylogarithmically large. This gives asymptotic solutions of longstanding conjectures of Chvátal and Lovász on Hamiltonicity of tough and vertex-transitive graphs.
Comments29 pages
