{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-graph-coloring",
  "semantic_family_id": "sf-graph-coloring",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "圖著色"
  },
  "summary": {
    "zh-Hant": "proper k-coloring使用至多k種顏色並使每條邊兩端顏色不同。"
  },
  "statement": {
    "zh-Hant": "完整核全部邊；非相鄰頂點可同色。一個proper coloring只給chromatic number上界；最優性需另有lower-bound證書，greedy不自動最優。"
  },
  "formula": {
    "tex": "C(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "C",
        "glyph": "C"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "e960d81a061905a155cac2e3e7ac2f524df62ae2f9f7f5a3437434b6ba160359"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無向簡單圖與每頂點color表。"
    },
    {
      "id": "a2",
      "zh-Hant": "顏色label重命名不改實質partition。"
    }
  ],
  "symbols": [
    {
      "token": "C",
      "role": "proper_vertex_coloring",
      "role_zh": "proper頂點著色",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "simple_graph",
      "role_zh": "有限圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "有限圖",
      "reason_zh": "完整邊集合。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "coloring為V到colors總映射。"
    },
    {
      "id": "mko-natural-number",
      "title_zh": "自然數",
      "reason_zh": "k作顏色數上界。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "每條邊兩端異色。",
      "使用色數不超過k。",
      "上界不等於chromatic optimum。"
    ],
    "common_misconception_zh": "要求所有頂點異色，或把greedy色數當最小色數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核指定coloring與上界；不宣稱最優除非有lower bound。",
    "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chromatic number與clique lower bound。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-graph-coloring.md 整理；2026-10-14 脈衝 R05（使用者手動接續 2026-10-31 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"
  }
}