{
  "abstract": "Fix an integer $r\\ge 3$. Let $K_r$ denote the complete graph on $r$ vertices and $t_{r-1}(n)$ the Turán number for $K_r$. For each integer $k$ satisfying $0\\le k \\le \\binom{n}{2}-t_{r-1}(n)$, $\\rev{ϕ_r(n,k)}$ denotes the minimum, over all $n$-vertex graphs with $t_{r-1}(n)+k$ edges, of the maximum size of a $K_r$-packing \\rev{(a family of edge-disjoint copies of $K_r$)} in the graph. The problem of characterizing the behavior of $ϕ_r(n,k)$ has a long history and dates back to Erdős for the case $r=3$. \\rev{Győri} proved that $ϕ_r(n,k)=(1-o(1))k$ whenever $k=o(n^2)$. We characterize the range of $k$ for which $ϕ_r(n,k)=(1-o(1))k$, showing that this equality holds if and only if $k=o(n^2)$. Moreover, we obtain an \\rev{asymptotic} upper bound (which falls below $1$) for $ϕ_r(n,k)/k$ whenever $k/n^2$ converges to a constant $δ$ satisfying $0 < δ\\le \\frac{1}{2(r-1)}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41371",
  "author_total": 3,
  "authors": [
    {
      "name": "Hanzhi Bai"
    },
    {
      "name": "Fuhong Ma"
    },
    {
      "name": "Jin Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9mh4227bfag3w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41371",
  "fulltext_url": null,
  "id": "9mh4227bfag3w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41371",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41371",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9mh4227bfag3w.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.41371",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41371"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$K_r$-Packings in Graphs with $t_{r-1}(n)+k$ Edges",
  "title_text": "K_r-Packings in Graphs with t_r−1(n)+k Edges",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/9mh4227bfag3w",
  "venue": "arXiv",
  "versions": []
}