{
  "abstract": "Jiang and Zhu [J. Combin. Theory Ser. B, 137 (2019) 112--117] asked whether every $k$-choosable graph is $(km+1:m)$-choosable for every positive integer $m$. We answer this question negatively by constructing a graph with chromatic number and choice number $k$ that is not $(km+1:m)$-choosable for every $k\\ge3$ and $m\\ge10$. More generally, for integers $k\\ge3$, $m\\ge2$, and $t\\ge0$, satisfying $m>8t+ε$, where $ε\\in\\{0,1\\}$ is the parity of $m$, we construct such a graph that is not $(km+t:m)$-choosable. This also answers negatively the equivalent question of Xu and Zhu [SIAM J. Discrete Math., 36 (2022) 2585--2601] asking whether the strong fractional choice number of every graph is at most its choice number. This also indicates that, for each $k\\ge3$, the supremum of the strong fractional choice numbers of graphs with choice number $k$ is at least $k+1/8$. When $t=0$, our construction yields counterexamples to multiplication for every $k\\ge3$ and $m\\ge2$. In particular, it yields a $3$-choosable graph on $91$ vertices that is not $(6:2)$-choosable, answering a question of Dvořák, Hu and Sereni [ Adv. Comb. (2019), Paper No. 5].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03151",
  "author_total": 2,
  "authors": [
    {
      "name": "Xiaolan Hu"
    },
    {
      "name": "Jiao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/evgt5g7fqsvpe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03151",
  "fulltext_url": null,
  "id": "evgt5g7fqsvpe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03151",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03151",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/evgt5g7fqsvpe.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03151",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03151"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Counterexamples to Relaxed Multiplication in List Coloring",
  "title_text": "Counterexamples to Relaxed Multiplication in List Coloring",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/evgt5g7fqsvpe",
  "venue": "arXiv",
  "versions": []
}