{
  "schema_version": "ocme-derivation-reader-v1",
  "guide_id": "derivation-mko-graph-coloring-01",
  "mko_id": "mko-graph-coloring",
  "title": {
    "zh-Hant": "完整著色與非最優greedy trace",
    "en": "A complete coloring and a nonoptimal greedy trace"
  },
  "contract": {
    "schema_version": "ocme-derivation-guide-v0.1",
    "guide_id": "derivation-mko-graph-coloring-01",
    "source": {
      "mko_id": "mko-graph-coloring",
      "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
      "sha256": "6c2084e53867b83ec1c3c9f3a21f1023e1d3b0910c9b23c6f774aa05a08ff54d",
      "english_sha256": "d5c203e54e6a16948a99c1802bfc693db9999eafa424314ef0841cca900e9476"
    },
    "mode": "worked_example",
    "audience_prerequisites": [
      "work"
    ],
    "goal": {
      "zh-Hant": "逐邊驗proper coloring並展示上界非自動最優。",
      "en": "Audit a proper coloring edgewise and show an upper bound need not be optimal."
    },
    "givens": [
      {
        "id": "G1",
        "statement": {
          "zh-Hant": "V={0,1,2,3}，E={{0,1},{1,2},{2,3}}；c=(1,2,1,2)。greedy依次處理0、3、1、2，取最小未被已著色鄰點使用的正色號。",
          "en": "V={0,1,2,3}, E={{0,1},{1,2},{2,3}}, c=(1,2,1,2). Greedy processes0,3,1,2 and uses the least positive color absent from colored neighbors."
        }
      }
    ],
    "steps": [
      {
        "id": "S01",
        "uses": [
          "G1",
          "work"
        ],
        "before_tex": "c:V\\to\\{1,2\\}",
        "after_tex": "c(0)=1,c(1)=2,c(2)=1,c(3)=2",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "四頂點各有一色，total table完整。",
          "en": "Each of the four vertices has one color, giving a complete total table."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "work"
          ],
          "explanation": {
            "zh-Hant": "color label本身可重命名。",
            "en": "Color labels themselves may be renamed."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S02",
        "uses": [
          "G1",
          "S01",
          "work"
        ],
        "before_tex": "\\{0,1\\}\\in E",
        "after_tex": "c(0)=1\\ne2=c(1)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "核第一邊。",
          "en": "Audit the first edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S01",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不能只核大部分邊。",
            "en": "Checking most edges is insufficient."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S03",
        "uses": [
          "G1",
          "S02",
          "work"
        ],
        "before_tex": "\\{1,2\\}\\in E",
        "after_tex": "c(1)=2\\ne1=c(2)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "核第二邊。",
          "en": "Audit the second edge."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S02",
            "work"
          ],
          "explanation": {
            "zh-Hant": "0與2同色合法，因無邊。",
            "en": "0 and 2 may share a color because they have no edge."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S04",
        "uses": [
          "G1",
          "S03",
          "work"
        ],
        "before_tex": "\\{2,3\\}\\in E",
        "after_tex": "c(2)=1\\ne2=c(3)",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "核第三邊，已遍歷全部E。",
          "en": "Audit the third edge, exhausting E."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S03",
            "work"
          ],
          "explanation": {
            "zh-Hant": "沒有要求非相鄰1與3異色。",
            "en": "Nonadjacent 1 and 3 need not have different colors."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S05",
        "uses": [
          "G1",
          "S04",
          "mko_definition"
        ],
        "before_tex": "\\text{all three edges proper},\\quad|c(V)|=2",
        "after_tex": "\\chi(G)\\le2",
        "relation": "implies",
        "rule": "mko_definition",
        "conditions": [],
        "explanation": {
          "zh-Hant": "一個proper coloring給兩色上界。",
          "en": "One proper coloring gives an upper bound of two."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S04",
            "mko_definition"
          ],
          "explanation": {
            "zh-Hant": "這一步不是一般由見證得到等號的規則。",
            "en": "This step does not generally turn a witness into equality."
          }
        },
        "leaf_basis": "mko_definition"
      },
      {
        "id": "S06",
        "uses": [
          "G1",
          "S05",
          "work"
        ],
        "before_tex": "\\text{greedy order }0,3,1,2",
        "after_tex": "g(0)=1,\\ g(3)=1,\\ g(1)=2,\\ g(2)=3",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "前兩點無已著色鄰點；1避0的1；2同時避1的2與3的1。",
          "en": "The first two vertices have no previously colored neighbors. Vertex 1 avoids color 1 at vertex 0. Vertex 2 avoids color 2 at vertex 1 and color 1 at vertex 3."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S05",
            "work"
          ],
          "explanation": {
            "zh-Hant": "這是完整四步trace，不只報最後色數。",
            "en": "This is a complete four-step trace, not just a color count."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S07",
        "uses": [
          "G1",
          "S06",
          "work"
        ],
        "before_tex": "g=(1,2,3,1)",
        "after_tex": "1\\ne2,\\quad2\\ne3,\\quad3\\ne1\\implies g\\text{ proper with3 colors}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "依三條原邊重驗greedy結果合法。",
          "en": "Reaudit the three original edges to validate the greedy result."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S06",
            "work"
          ],
          "explanation": {
            "zh-Hant": "合法不代表最少，因c已用兩色。",
            "en": "Validity does not mean minimum, since c uses two colors."
          }
        },
        "leaf_basis": "work"
      },
      {
        "id": "S08",
        "uses": [
          "G1",
          "S07",
          "work"
        ],
        "before_tex": "|g(V)|=3>2=|c(V)|",
        "after_tex": "g\\text{ is not a minimum-color solution}",
        "relation": "implies",
        "rule": "work",
        "conditions": [],
        "explanation": {
          "zh-Hant": "兩色見證明確否定此greedy結果的最優性。",
          "en": "The two-color witness explicitly refutes optimality of this greedy result."
        },
        "reverse_audit": {
          "kind": "premise_audit",
          "required_ids": [
            "G1",
            "S07",
            "work"
          ],
          "explanation": {
            "zh-Hant": "不由一個order推所有greedy都失敗。",
            "en": "This one order does not imply that every greedy execution fails."
          }
        },
        "leaf_basis": "work"
      }
    ],
    "result": {
      "zh-Hant": "c為proper兩色上界；指定greedy用三色而非最優。",
      "en": "c supplies a proper two-coloring and hence an upper bound; the specified greedy run uses three colors and is not optimal."
    },
    "negative_controls": [
      {
        "claim": {
          "zh-Hant": "非相鄰0、2必須異色。",
          "en": "Nonadjacent 0,2 must differ in color."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "它們在c同色且全部邊合法。",
          "en": "They share a color in c while every edge is proper."
        }
      },
      {
        "claim": {
          "zh-Hant": "greedy三色即chi=3。",
          "en": "Greedy using three colors gives chi=3."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "c是明示兩色見證。",
          "en": "c is an explicit two-color witness."
        }
      },
      {
        "claim": {
          "zh-Hant": "只核邊01與12即可。",
          "en": "Checking01 and 12 suffices."
        },
        "expected": "reject",
        "reason": {
          "zh-Hant": "邊23也有獨立異色義務。",
          "en": "Edge 23 has its own distinct-color obligation."
        }
      }
    ],
    "status": "candidate",
    "gaps": [],
    "verification": {
      "structure": "checked_local_script",
      "mathematics": "candidate_argument",
      "formalization": "not_formalized"
    },
    "rule_glossary": [
      {
        "id": "mko_definition",
        "kind": "cited",
        "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。",
          "en": "Check every edge in the fully specified graph. Nonadjacent vertices may share a color. One proper coloring supplies only a chromatic-number upper bound; optimality needs a separate lower-bound certificate, and a greedy result is not automatically optimal."
        },
        "source": {
          "mko_id": "mko-graph-coloring",
          "sha256": "6c2084e53867b83ec1c3c9f3a21f1023e1d3b0910c9b23c6f774aa05a08ff54d",
          "commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f"
        }
      },
      {
        "id": "work",
        "kind": "prerequisite",
        "statement": {
          "zh-Hant": "有限無向簡單圖的邊為無loop、無重複unordered pairs；圖性質須對完整明示V、E逐一檢查。 proper coloring為total頂點色表且每邊兩端異色；使用色數是像集大小；χ為所有proper colorings的最小使用色數。",
          "en": "A finite simple undirected graph has loop-free duplicate-free unordered edges; graph properties are audited on all declared V and E. A proper coloring is a total vertex-color table with distinct endpoint colors on every edge; its color count is its image size; chi is the minimum color count over all proper colorings."
        }
      }
    ],
    "revision_log": [],
    "numeric_checks": [
      {
        "id": "N01",
        "lhs": "3",
        "rhs": "2",
        "relation": ">"
      },
      {
        "id": "N02",
        "lhs": "2",
        "rhs": "2"
      }
    ]
  },
  "presentation": {
    "md_scope": {
      "zh-Hant": "只核明示colorings與greedy執行，不作一般最優保證。",
      "en": "Only supplied colorings and one greedy execution, with no general optimality guarantee."
    },
    "md_symbols": [
      [
        "V,E",
        "完整頂點與邊集合",
        "complete vertex and edge sets"
      ],
      [
        "c,g",
        "兩個coloring",
        "the two colorings"
      ]
    ],
    "md_error_controls": [
      {
        "title": "all-edge / All edges",
        "zh": "明列三條邊的端點色號。",
        "en": "List endpoint colors for all three edges."
      },
      {
        "title": "greedy順序 / Greedy order",
        "zh": "0、3、1、2不是頂點自然順序。",
        "en": "The order0,3,1,2 differs from natural vertex order."
      }
    ],
    "md_open_items": []
  },
  "revision": {
    "id": "ocme-new142-reader-adoption-20261021",
    "collection": "coverage-20261021",
    "source_commit": "b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f",
    "adopted_source_sha256": "bd902ee022f3e9a7e028bcbc16fe7a15bfc5763b4e962ada50bb0848119b4213",
    "original_source_sha256": "77038d6e68dcdd481876b761a977fba847b3a721ba29cab154d3c196492bc827",
    "owner_adopted": true,
    "bilingual_sync": "complete_source_bound",
    "formal_Lean": "NOT_RUN",
    "reader_review": "142_of_142_source_bound",
    "wording_review": "original_Lead_decisions_resolved",
    "glossary_reader_paraphrases": [
      "mko_definition"
    ],
    "independent_AI_concurrence": "SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW",
    "issue_ids": [
      "BR142-S032",
      "BR142-S033",
      "BR142-S034",
      "BR142-P009",
      "BR142-P007",
      "BR142-S030",
      "BR142-S031",
      "BR142-P008"
    ],
    "parent_manifest_sha256": "d29f4cae8229ebd24f60d496cb1d3cd4ad8c0a4dca9d44daab0c3245c544283a",
    "scope": "Owner-adopted reader revision. Lead and bilingual review concern the recorded text and premises; no new universal or Lean proof, canonical promotion or historical-label rewrite."
  }
}