{
  "abstract": "We construct trees with perfect matchings and independence sequences that fail log-concavity near the independence number $α$. For every integer $k\\geq1$, one family fails at $α-k$ and has a witness of order $O(k^2\\log(k+2))$. For every integer $t\\geq2$, a single tree has simultaneous failures at $α-1,α-3,\\ldots,α-(2t-1)$, with a witness of order $O(t^3)$. All these trees satisfy $α=N/2$, where $N$ is the number of vertices, and contradict Conjecture 4.2 of Ramos and Sun. A sufficient condition uses a fixed tree and the numbers of attached leaves. For fixed $k$ or $t$, explicit bounds give geometric decay of $a_j^2/(a_{j-1}a_{j+1})$ at these indices as the trees grow; $a_j$ counts independent $j$-sets. The proofs count independent sets by empty matching pairs. The Lean proof assistant checks formal proofs of the parameterized results and their quantitative bounds; exact 90- and 180-vertex examples illustrate the constructions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18340",
  "author_total": 1,
  "authors": [
    {
      "name": "Heshun Fan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/78b2e77359jx2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18340",
  "fulltext_url": null,
  "id": "78b2e77359jx2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18340",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LO"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7221750959",
  "pdf_url": "https://arxiv.org/pdf/2610.18340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/78b2e77359jx2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:20Z",
      "native_id": "2610.18340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Prescribed and simultaneous failures of log-concavity in trees with perfect matchings",
  "title_text": "Prescribed and simultaneous failures of log-concavity in trees with perfect matchings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:20Z",
  "url": "https://research.draconitic.com/paper/78b2e77359jx2",
  "venue": "arXiv",
  "versions": []
}