{
  "abstract": "Let $C_n^r$ denote the direct sum of $r$ cyclic groups of order $n$, where $n\\ge2$ and $r\\ge1$. Properties C and D predict a uniform multiplicity structure for extremal zero-sum problems over $C_n^r$: every sequence of maximum length avoiding nonempty zero-sum subsequences of length at most $n$, or of length exactly $n$, respectively, should have the form $T^{n-1}$. We prove that, for every $r\\ge4$ and every even $n\\ge2$, each of these properties holds if and only if $n$ is a power of two. This gives negative answers to the general Property C problem posed by Gao, Hou, Schmid and Thangadurai Open Problems 1.2{GHST} in 2007, and disproves the Property D conjecture proposed by Gao and Geroldinger [Conjecture 7.2]{GG} in 2006.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25271",
  "author_total": 2,
  "authors": [
    {
      "name": "Guoqing Wang"
    },
    {
      "name": "Xiangneng Zeng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0hxg481na97z3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25271",
  "fulltext_url": null,
  "id": "0hxg481na97z3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25271",
  "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"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223043247",
  "pdf_url": "https://arxiv.org/pdf/2610.25271",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0hxg481na97z3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.25271",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25271"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Inverse Zero-Sum Problems of Erdős--Ginzburg--Ziv constant: Property C and Property D conjectures",
  "title_text": "Inverse Zero-Sum Problems of Erdős--Ginzburg--Ziv constant: Property C and Property D conjectures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/0hxg481na97z3",
  "venue": "arXiv",
  "versions": []
}