{
  "abstract": "Garamvölgyi, Jordán, Király and Villányi [{{\\bf Forum Math. Pi} \\textbf{13} (2025), Paper No.~e11}] posed two sharp connectivity conjectures for packing rigid spanning subgraphs: one for the equal-dimensional case and the other for the packing of a $d$-rigid spanning subgraph with a spanning tree. We prove a unified theorem: for arbitrary positive integers $d_1,\\ldots,d_s$, every $\\sum_{i=1}^{s}d_i(d_i+1)$-connected graph contains pairwise edge-disjoint spanning subgraphs $H_1,\\ldots,H_s$ such that $H_i$ is $d_i$-rigid for every $i$. The connectivity bound is sharp whenever $\\sum_{i=1}^{s}d_i(d_i+1)\\ge4$. As special cases, the theorem settles both conjectures, confirms the conjecture of Garamvölgyi, Jordán and Király [{\\bf J. Combin. Theory Ser. B} \\textbf{166} (2024), 1--29] that every $tk(k+1)$-connected graph contains $t$ pairwise edge-disjoint $k$-connected spanning subgraphs, and gives the sharp threshold $d(d+1)+2r$ for packing one $d$-rigid spanning subgraph together with $r$ pairwise edge-disjoint spanning trees. We also obtain two upper bounds related to Thomassen's conjecture on highly connected orientations of graphs. If $f(q)$ is the least integer such that every $f(q)$-connected graph has a $q$-connected orientation, then $f(q)\\le(25q^2+41q-16)/2$ for every $q\\ge3$ and $f(q)\\le8q^2+212q+1404=(8+o(1))q^2$ for all sufficiently large $q$; these two results reduce the leading coefficient in the previous quadratic bound from $320$ to $25/2$ for every $q\\ge3$ and $8$ for all sufficiently large $q$. Compared to the bound for $f(q)$ obtained by Garamvölgyi et al. we obtain the better bounds, not only through our tight rigidity result but also by exploiting the leftover edges when we remove two edge-disjoint spanning (sufficiently) rigid graphs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24463",
  "author_total": 3,
  "authors": [
    {
      "name": "Hanzhi Bai"
    },
    {
      "name": "Jørgen Bang-Jensen"
    },
    {
      "name": "Jin Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/964w4kndk7gr9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24463",
  "fulltext_url": null,
  "id": "964w4kndk7gr9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24463",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214028877",
  "pdf_url": "https://arxiv.org/pdf/2609.24463",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/964w4kndk7gr9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24463",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24463"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sharp connectivity thresholds for mixed rigidity packings and improved bounds for highly connected orientations of graphs",
  "title_text": "Sharp connectivity thresholds for mixed rigidity packings and improved bounds for highly connected orientations of graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/964w4kndk7gr9",
  "venue": "arXiv",
  "versions": []
}