{
  "abstract": "We show that almost every graph admits a partition of its vertex set into three parts such that no two adjacent vertices have the same number of neighbors in each of the three parts. Equivalently, for $G\\sim G(n,1/2)$, $χ_m(G)\\le3$ with high probability, improving the previously known bound of five. Here $χ_m(G)$ denotes the multiset chromatic number of $G$, the minimum number of parts in a vertex partition whose neighbor-count vectors distinguish every pair of adjacent vertices. In fact, the three-part bound holds for every fixed $0.185<p<0.509$. More generally, for every fixed $p\\in(0,1)$, $G\\sim G(n,p)$ satisfies $χ_m(G)\\le4$ with high probability. These results are obtained by converting the unresolved edges of a carefully chosen initial partition into hyperplanes of a Boolean cube and applying the Linial--Radhakrishnan theory of essential covers. We also determine how $χ_m(G)$ grows when the graph is polynomially close to complete. For every fixed $β\\in(0,1)$ and $G\\sim G\\!\\left(n,1-n^{-(1-β)}\\right)$, with high probability $\\frac{2}β\\le χ_m(G)\\le \\left\\lfloor\\frac{2}β\\right\\rfloor+5$. Thus $χ_m(G)=2/β+O(1)$. The lower bound is spectral, while the upper bound follows from multinomial anti-concentration and the Lov'asz Local Lemma.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23362",
  "author_total": 2,
  "authors": [
    {
      "name": "Arash Ahadi"
    },
    {
      "name": "Sharareh Alipour"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8x929qvq4d6ar.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23362",
  "fulltext_url": null,
  "id": "8x929qvq4d6ar",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23362",
  "language": null,
  "licence": "CC-BY-SA-4.0",
  "licence_label": "CC BY-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213999076",
  "pdf_url": "https://arxiv.org/pdf/2609.23362",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8x929qvq4d6ar.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23362",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23362"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Multiset Colorings of Random Graphs Across Density Regimes",
  "title_text": "Multiset Colorings of Random Graphs Across Density Regimes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/8x929qvq4d6ar",
  "venue": "arXiv",
  "versions": []
}