{
  "abstract": "Shelburne and van Willigenburg (arXiv:2604.26158) characterize the Schur-positive complete multipartite graphs and leave open whether the graphs~$G=K_{(3,\\,2^β)}$ are $e$-positive. We resolve this question and, together with their classification, characterize all $e$-positive complete multipartite graphs. Our main result is an explicit, manifestly nonnegative $e$-expansion of~$X_G$ whose coefficients are expressed in terms of the restricted-injection numbers. Our main idea is to derive a marker-variable coefficient-extraction formula for the $e$-coefficients of arbitrary complete multipartite graphs from the elementary--monomial Cauchy identity. For the particular graph~$G$, this formula reduces the proof to three coefficient families, which we evaluate using Dickson polynomials and recurrences for these numbers.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.19110",
  "author_total": 3,
  "authors": [
    {
      "name": "Ariel Y. Sun"
    },
    {
      "name": "David G. L. Wang"
    },
    {
      "name": "Watson Z. Y. Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c84wydtedap25.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.19110",
  "fulltext_url": null,
  "id": "c84wydtedap25",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.19110",
  "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",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7170064528",
  "pdf_url": "https://arxiv.org/pdf/2607.19110",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c84wydtedap25.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2607.19110",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.19110"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An $e$-positive classification for complete multipartite graphs",
  "title_text": "An e-positive classification for complete multipartite graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/c84wydtedap25",
  "venue": "arXiv",
  "versions": []
}