{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-chromatic-number",
  "semantic_family_id": "sf-chromatic-number",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "色數"
  },
  "summary": {
    "zh-Hant": "有限簡單圖的色數是proper頂點著色所需顏色數的最小值，需同時具備上界與下界證書。"
  },
  "statement": {
    "zh-Hant": "一個proper k-coloring只證chi(G)<=k；等號還須排除(k-1)-coloring。小型完整圖以exhaustive search給下界並用明示著色給上界；greedy結果不當作最優性。空頂點圖採chi=0，非空無邊圖採chi=1。"
  },
  "formula": {
    "tex": "chi(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>chi</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "chi",
        "glyph": "chi"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "8472ba0a294de948509b13c6a8352eb99cf23836697123f599787b9ead4d512c"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無loop簡單無向圖，頂點標號0至n-1。"
    },
    {
      "id": "a2",
      "zh-Hant": "exact-number題限定n<=9並完整搜尋所有較少顏色配置。"
    }
  ],
  "symbols": [
    {
      "token": "chi",
      "role": "chromatic_number",
      "role_zh": "色數",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "finite_simple_graph",
      "role_zh": "有限簡單圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph-coloring",
      "title_zh": "圖著色",
      "reason_zh": "proper coloring與顏色配置。"
    },
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "最小顏色數。"
    },
    {
      "id": "mko-optimization-problem",
      "title_zh": "最佳化問題",
      "reason_zh": "上界與最小性責任。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "k-colorable表示最多k色，不表示恰需k色。",
      "等號由constructive upper bound與valid lower bound合成。",
      "greedy著色只是一個上界演算法。"
    ],
    "common_misconception_zh": "找到proper k-coloring便直接宣稱chi(G)=k。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "exact搜尋只覆蓋完整明示的小型有限圖；不作一般複雜度或近似保證。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入perfect graph與chromatic polynomial。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-chromatic-number.md 整理；2026-10-14 脈衝 R01（使用者手動接續 2026-11-06 工作序列；實際日期仍為 2026-10-14），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-14"
  }
}