{"schema_version":"ocme-site-results-v1","locale":"zh-Hant","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-graph"},"kind":"guides","total":5,"pages":1,"page":1,"entries":[{"id":"derivation-mko-graph-01","title":{"zh-Hant":"有限簡單無向圖：鄰居、度數與握手公式","en":"Finite simple undirected graphs: neighbours, degrees and the handshake formula"},"summary":{"zh-Hant":"設 V={1,2,3,4,5}，邊以端點對清單輸入：(1,2)、(2,1)、(2,3)、(3,4)、(2,4)、(4,2)，每一項表示一條無向邊。把輸入正規化為邊集合 E，求每個頂點的鄰居集合 N(v) 與度數 deg(v)，核對度數和等於 2|E|，並對每個有限簡單無向圖證明握手公式 Σ_{v∈V} deg(v)=2|E|。","en":"Let V={1,2,3,4,5}, with edges entered as a list of endpoint pairs: (1,2), (2,1), (2,3), (3,4), (2,4), (4,2), each entry denoting an undirected edge. Normalize the input to the edge set E, find the neighbour set N(v) and degree deg(v) of every vertex, check that the degree sum equals 2|E|, and prove the handshake formula Σ_{v∈V} deg(v)=2|E| for every finite simple undirected graph."},"mko_ids":["mko-graph"]},{"id":"derivation-mko-graph-isomorphism-01","title":{"zh-Hant":"圖同構：以完整映射表逐項核對雙射與鄰接當且僅當","en":"Graph isomorphism: checking bijectivity and both directions of adjacency preservation from a complete mapping table"},"summary":{"zh-Hant":"G 的頂點為 1、2、3、4，邊為 {1,2}、{2,3}、{3,4}、{1,4}；H 的頂點為 a、b、c、d，邊為 {a,c}、{b,c}、{b,d}、{a,d}。證明映射 f：1↦a、2↦c、3↦b、4↦d 是圖同構 G→H；說明雙射 g：1↦a、2↦b、3↦c、4↦d 不是同構；並證明路徑 P（邊 {1,2}、{2,3}、{3,4}）與 H 不同構，即使 f 把 P 的每條邊都送到 H 的邊。","en":"G has vertices 1, 2, 3, 4 and edges {1,2}, {2,3}, {3,4}, {1,4}; H has vertices a, b, c, d and edges {a,c}, {b,c}, {b,d}, {a,d}. Prove that f: 1↦a, 2↦c, 3↦b, 4↦d is a graph isomorphism G→H; show that the bijection g: 1↦a, 2↦b, 3↦c, 4↦d is not an isomorphism; and prove that the path P (edges {1,2}, {2,3}, {3,4}) is not isomorphic to H, even though f sends every edge of P to an edge of H."},"mko_ids":["mko-graph-isomorphism"]},{"id":"derivation-mko-graph-walk-simple-path-01","title":{"zh-Hant":"圖的 walk 與 simple path：逐對核對、長度與 trail 的區別","en":"Graph walks and simple paths: pair-by-pair checks, length and the difference from trails"},"summary":{"zh-Hant":"有限無向無 loop 簡單圖 G：V={0,1,2,3,4}，E={{0,1},{1,2},{2,3},{0,3},{1,3},{3,4}}。對頂點序列 W₁=(0,1,2,3,4)、W₂=(0,1,3,1,2)、W₃=(0,2,3)、W₄=(2,3,0,1,3,4)、W₅=(4)、W₆=(1,1) 逐一判定是否為 walk、長度為何、是否為 simple path；並證明 simple path 不重複邊，以及本圖中 simple path 的長度至多為 4。","en":"Finite undirected loop-free simple graph G: V={0,1,2,3,4}, E={{0,1},{1,2},{2,3},{0,3},{1,3},{3,4}}. For the vertex sequences W₁=(0,1,2,3,4), W₂=(0,1,3,1,2), W₃=(0,2,3), W₄=(2,3,0,1,3,4), W₅=(4) and W₆=(1,1), decide whether each is a walk, its length, and whether it is a simple path; also prove that a simple path repeats no edge and that in this graph a simple path has length at most 4."},"mko_ids":["mko-graph-walk-simple-path"]},{"id":"derivation-mko-graph-coloring-01","title":{"zh-Hant":"完整著色與非最優greedy trace","en":"A complete coloring and a nonoptimal greedy trace"},"summary":{"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."},"mko_ids":["mko-graph-coloring"]},{"id":"derivation-mko-graph-matching-01","title":{"zh-Hant":"P4中maximal與maximum的分界","en":"Maximal versus maximum in P4"},"summary":{"zh-Hant":"完整核選邊與端點，再分清三種匹配性質。","en":"Audit selected edges and endpoints, then distinguish matching properties."},"mko_ids":["mko-graph-matching"]}],"title":"概念推導導讀"}