{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30165",
  "author_total": 3,
  "authors": [
    {
      "name": "Micha Christoph"
    },
    {
      "name": "Zach Hunter"
    },
    {
      "name": "Benny Sudakov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1mdxednkgv5dt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30165",
  "fulltext_url": null,
  "id": "1mdxednkgv5dt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30165",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7214297960",
  "pdf_url": "https://arxiv.org/pdf/2609.30165",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1mdxednkgv5dt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30165",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30165"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Thinning and sprinkling: from robust sampling to almost Hamiltonicity",
  "title_text": "Thinning and sprinkling: from robust sampling to almost Hamiltonicity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1mdxednkgv5dt",
  "venue": "arXiv",
  "versions": []
}