{
  "abstract": "For every fixed $k\\ge 3$, every finite nonempty clutter $\\mathcal{H}$ can be realized exactly as the family of inclusion-minimal terminal traces of the $k$-bad odd cycles, namely the odd cycles $C$ satisfying $χ(G[V(C)])\\ge k+1$. The host graph $G$ may be chosen $2$-connected, with $T=V(\\mathcal{H})$ independent, and with $χ(G)=k+1$, $ω(G)=k$, and $|E(G)|\\le k|V(G)|$. Given any integer $B\\ge 1$, one realization simultaneously preserves the transversal number, the fractional packing number, and every integer $c$-packing number for $1\\le c\\le B$. Thus chromatic richness on odd-cycle spans has the full packing-covering complexity of arbitrary finite set systems, even in sparse graphs whose chromatic number exceeds their clique number by one. As a consequence, for every $b,N\\ge 1$ and every $\\varepsilon>0$, there is such a graph with $τ_k(G)=N$, $ν_k^{1/c}(G)=c$ for every $1\\le c\\le b$, and $ν_k^*(G)<1+\\varepsilon$. Hence every finite-congestion and fractional Erdős--Pósa property fails for $k\\ge 3$, with an asymptotically optimal fractional obstruction. This contrasts sharply with $k=2$, where the relevant cycles are the ordinary odd cycles: the integral property fails, while Reed's theorem yields every congestion level at least two and the fractional property. We also characterize the shortest $k$-bad odd cycles by a single fixed witness graph.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26158",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuyan Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/btkqy73pcm43m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26158",
  "fulltext_url": null,
  "id": "btkqy73pcm43m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26158",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214118173",
  "pdf_url": "https://arxiv.org/pdf/2609.26158",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/btkqy73pcm43m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.26158",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26158"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Hypergraph Universality and Erdos-Posa Failure for Chromatically Rich Odd Cycles",
  "title_text": "Hypergraph Universality and Erdos-Posa Failure for Chromatically Rich Odd Cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/btkqy73pcm43m",
  "venue": "arXiv",
  "versions": []
}