{
  "abstract": "We study the ground states of the antiferromagnetic three-state Potts model in which the third state carries a single-ion energy cost $D_3$, on arbitrary lattices, and show that the difficulty of the ground-state problem is controlled by the maximum coordination $Δ$ alone; on planar lattices with $Δ\\le5$ it is classified completely. On lattices with coordination at most three the third state heals frustrated bonds one for one: every graph of maximum degree three other than $K_4$ has an independent set of sites that meets every odd cycle and whose size equals the number $\\operatorname{fr}$ of frustrated bonds of the Ising ground state. This identity also follows from known results on the chromatic vertex stability of graphs; we give a constructive proof. Hence $E_0 (D_3)=\\min(D_3,1) \\operatorname{fr}$ for all $D_3$, computable in polynomial time on planar lattices. In graph-theoretic terms, independent odd cycle transversal is solvable in polynomial time on planar subcubic graphs. On planar lattices with maximum coordination four, in contrast, deciding the ground-state energy is NP-complete for every fixed $0\\le D_3<2$ and polynomial for $D_3\\ge2$. The boundary is set by a local principle: a site of coordination $d$ in the third state relieves at most $\\lfloor d/2\\rfloor$ frustrated bonds, so the third state is useless once $D_3\\ge\\lfloorΔ/2\\rfloor$; this threshold is sharp for every $Δ\\ge3$ and reproduces the triangular-lattice point $D_3^*=3$ of the equivalent spin-1 Blume--Capel antiferromagnet. From a single Ising ground state the proof builds an exact ground state in polynomial time, which for quasicrystalline, foam-like and defective lattices with thousands of sites takes seconds to minutes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.41277",
  "author_total": 5,
  "authors": [
    {
      "name": "Jeehoon Choi"
    },
    {
      "name": "Suhwan Yoo"
    },
    {
      "name": "Chaeyeon Kim"
    },
    {
      "name": "Hee Young Kwon"
    },
    {
      "name": "Changyeon Won"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3kpxch34wrmkb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.41277",
  "fulltext_url": null,
  "id": "3kpxch34wrmkb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.41277",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.41277",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3kpxch34wrmkb.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:15Z",
      "native_id": "2610.41277",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.41277"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Coordination three heals, coordination four is hard: the antiferromagnetic Potts model with a costly third state",
  "title_text": "Coordination three heals, coordination four is hard: the antiferromagnetic Potts model with a costly third state",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:52Z",
  "url": "https://research.draconitic.com/paper/3kpxch34wrmkb",
  "venue": "arXiv",
  "versions": []
}