{
  "abstract": "We investigate the $L_2$ Turán problems for various small directed graphs, specifically focusing on self-converse tournaments and stability versions. First, we determine the exact maximum $L_2$ norm squared of the out-degree sequence for digraphs avoiding the transitive tournament $TT_4$ and the strongly connected tournament $R_4$, answering open questions from recent paper. We prove that the complete directed 3-partite Turán graph $T_3(m)$ exactly maximizes the $L_2$ norm squared for $TT_4$-free digraphs. For $R_4$-free digraphs, the maximum is achieved by $T_3(m)$ except when $m \\equiv 1 \\pmod 3$, where peeling off a terminal sink vertex to form $T_3(m-1) \\to v$ strictly increases the objective. We complement these results with exact values and a conjecture for the regular tournament $Reg_5$. Furthermore, we prove a stability version for $\\vec{C}_3$-free digraphs: any sequence of digraphs asymptotically achieving the maximum $L_2$ density must have an edit distance of $O(δ^{1/2})m^2$ to the extremal ordered digon-chain $\\vec{F}_{m,2}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05042",
  "author_total": 1,
  "authors": [
    {
      "name": "Daniel Iľkovič"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fw70xm4etsbfr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05042",
  "fulltext_url": null,
  "id": "fw70xm4etsbfr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05042",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7211917456",
  "pdf_url": "https://arxiv.org/pdf/2609.05042",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fw70xm4etsbfr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:18Z",
      "native_id": "2609.05042",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05042"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "$L_2$ Turán Problems for Small Tournaments and Stability",
  "title_text": "L_2 Turán Problems for Small Tournaments and Stability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:18Z",
  "url": "https://research.draconitic.com/paper/fw70xm4etsbfr",
  "venue": "arXiv",
  "versions": []
}