{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-bipartite-graph",
  "semantic_family_id": "sf-bipartite-graph",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "二分圖"
  },
  "summary": {
    "zh-Hant": "有限無向簡單圖若全體頂點可分為兩個互斥部分且每條邊跨部，則為二分圖。"
  },
  "statement": {
    "zh-Hant": "核完整頂點cover、兩部互斥及全部邊跨部。失敗partition不證非二分；奇cycle可作否證。孤立頂點可置任一部，空部是否允許須明示。"
  },
  "formula": {
    "tex": "B(G)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>B</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "B",
        "glyph": "B"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "G",
          "glyph": "G"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "f5643dd8ec67109d5989c6ad78fc2d8a5a35444a58cf3e49b79f06dcaf481e91"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "完整有限無向無loop簡單圖。"
    },
    {
      "id": "a2",
      "zh-Hant": "本輪允許無邊圖的一部為空；所有頂點仍須被cover。"
    }
  ],
  "symbols": [
    {
      "token": "B",
      "role": "bipartite_graph",
      "role_zh": "二分圖",
      "scope": "representative_formula"
    },
    {
      "token": "G",
      "role": "simple_graph",
      "role_zh": "有限簡單圖",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-graph",
      "title_zh": "有限圖",
      "reason_zh": "完整頂點邊集合。"
    },
    {
      "id": "mko-subset",
      "title_zh": "子集合",
      "reason_zh": "兩部為頂點子集。"
    },
    {
      "id": "mko-function-mapping",
      "title_zh": "映射",
      "reason_zh": "二著色等價表示。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "兩部互斥且聯集為V。",
      "每條邊兩端不同部。",
      "奇cycle是非二分證書；單一partition失敗不是。"
    ],
    "common_misconception_zh": "漏掉孤立點、由一個失敗partition判非二分，或要求非相鄰點顏色不同。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只處理有限無向簡單圖；不外推有向圖。",
    "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matching與二分圖定理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-bipartite-graph.md 整理；2026-10-14 脈衝 R02（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"
  }
}