{
  "abstract": "We determine a sharp threshold for the adaptable 2-colorability of a random graph equipped with a uniformly random, not necessarily proper, red/blue coloring of the edges. To accomplish this, we characterize a family of subgraphs along with edge colorings whose inclusion or exclusion determines adaptable $2$-colorability. We further show that above the threshold, a long path with alternating edge colors is formed. We use this path to prove the existence of such a subgraph in the supercritical regime. We then provide and prove symmetric bounds on the critical window for $2$-adaptable colorability. Particularly, we prove bounds matching that of the critical windows for the giant component in the Erd$ő$s-R$é$nyi random graph model as well as the satisfiability of a random $2$-SAT instance. Finally, we show that below the critical window, the solution space of adaptable $2$-colorings remains connected, that is one can travel from one adaptable $2$-coloring to another by a sequence of $2$-colorings which differ on $O(\\log{n})$ many vertices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12214",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Snow"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cq5rxzghvgsrk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12214",
  "fulltext_url": null,
  "id": "cq5rxzghvgsrk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12214",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7212809974",
  "pdf_url": "https://arxiv.org/pdf/2609.12214",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cq5rxzghvgsrk.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.12214",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12214"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Critical Window for Adaptable 2-Colorability",
  "title_text": "On the Critical Window for Adaptable 2-Colorability",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/cq5rxzghvgsrk",
  "venue": "arXiv",
  "versions": []
}