{
  "abstract": "We study the proper hat guessing game on graphs, introduced by Adriaensen et al. in Hat guessing with proper colorings. In this game, the players are seated on the vertices of a graph $G$ and assigned hats from a set of $k$ colors such that the resulting assignment forms a proper coloring. The visibility of each vertex is limited to the hat colors of their neighborhood. Then they must simultaneously output a guess about the color of their own hat. The players win if at least one guess is correct. A parameter related to this problem is the proper hat guessing number $\\operatorname{HG}_{P}(G)$ that is the maximum number of colors $m$ such that the players can guarantee a winning strategy. Motivated by the work of Shurman et al. \\cite{shurman2026upper}, we establish the first upper bound that depends both on the number of vertices $n$ and the maximum degree $Δ$ in the case where $Δ\\geq \\frac{n}{e+1}$. This result leads us to show that the proper hat guessing number of the binomial random graph $G_{n,1/2}$ is bounded above by $cn$, where $c \\approx 1.366$. Finally, we prove that graphs of maximum degree $(1-γ)n$ for some fixed $γ\\in (0,1]$ cannot have $\\operatorname{HG}_{P}(G) = (2-o(1))n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.04124",
  "author_total": 1,
  "authors": [
    {
      "name": "Ioannis Kakatelis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4152nn7d73qev.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.04124",
  "fulltext_url": null,
  "id": "4152nn7d73qev",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.04124",
  "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.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7220739317",
  "pdf_url": "https://arxiv.org/pdf/2610.04124",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4152nn7d73qev.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.04124",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.04124"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "An upper bound on the proper hat guessing number of graphs",
  "title_text": "An upper bound on the proper hat guessing number of graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/4152nn7d73qev",
  "venue": "arXiv",
  "versions": []
}