{
  "abstract": "Let $G$ be an $n$-vertex graph and let $\\ell:V(G)\\to\\mathbb{F}_2$ prescribe degree parities. A set $S\\subseteq V(G)$ is $\\ell$-admissible if every $v\\in S$ has degree congruent to $\\ell(v)$ modulo $2$ in $G[S]$. Let $h_\\ell(G)$ be the maximum order of an $\\ell$-admissible set, set $f_{\\mathrm{oe}}(G):=\\min_\\ell h_\\ell(G)$, and write $f_o(G):=h_{\\mathbf{1}}(G)$, where $\\mathbf{1}(v)=1$ for every $v\\in V(G).$ We prove three main results for graphs without isolated vertices. First, by extending Zeng's odd-cut method to arbitrary parity prescriptions and introducing a one-sided completion lemma, we show that $h_\\ell(G)\\ge 2n/11$ for every $\\ell$. Consequently, $f_{\\mathrm{oe}}(G)\\ge 2n/11$, improving the previous bound $2n/21$. Second, for bipartite graphs we derive lower bounds on $f_o(G)$ in terms of the $\\mathbb{F}_2$-rank of the bipartite adjacency matrix and combine them to obtain \\[ f_o(G)\\ge \\left(\\frac14+\\frac1{256}\\right)n=\\frac{65}{256}n. \\] Thus, in the bipartite case, the factor $2$ in Scott's bound $f_o(G)\\ge n/(2χ(G))$ can be replaced by $128/65<2$. Finally, a fourth-moment argument gives, for $α=α(G)\\ge2$ and $δ(G)\\ge 1$, \\[ f_o(G)\\ge \\fracα{2}+\\frac{\\log_3α}{8} -\\frac14\\log_3\\log_3\\sqrtα. \\] We also construct bipartite graphs satisfying \\[ f_o(G)\\le \\frac{α(G)}2+\\log_2\\!\\bigl(α(G)+1\\bigr)+\\frac12, \\] showing that the logarithmic additive improvement over Scott's bound $f_o(G)\\geα(G)/2$ has the optimal order of magnitude.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.02339",
  "author_total": 6,
  "authors": [
    {
      "name": "Qiwen Guo"
    },
    {
      "name": "Gregory Gutin"
    },
    {
      "name": "Yiming Hao"
    },
    {
      "name": "Yongtang Shi"
    },
    {
      "name": "Yong Zhang"
    },
    {
      "name": "Yacong Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0azpsgdp1a4ge.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.02339",
  "fulltext_url": null,
  "id": "0azpsgdp1a4ge",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.02339",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7172428030",
  "pdf_url": "https://arxiv.org/pdf/2608.02339",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0azpsgdp1a4ge.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2608.02339",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.02339"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounds on Odd and Odd-Even Induced Subgraphs",
  "title_text": "Bounds on Odd and Odd-Even Induced Subgraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0azpsgdp1a4ge",
  "venue": "arXiv",
  "versions": []
}