第 18967 題：Euler trail與circuit

原題與已知條件

完整有限簡單無向圖、edge support與明示walk。操作=euler-trail-audit；資料={"n":19,"edges":[[0,1],[0,18],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

• G001 — 完整有限簡單無向圖、edge support與明示walk。操作=euler-trail-audit；資料={"n":19,"edges":[[0,1],[0,18],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11],[11,12],[12,13],[13,14],[14,15],[15,16],[16,17],[17,18]],"vertex_walk":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。

定義與適用範圍

沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。

• mko-eulerian-trail

逐步推導

source operation=euler-trail-audit;original snapshot and mathematics retained

explicit ordered source output fields=[vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary]

complete ORIGINAL carrier Fin(19);canonical ORIGINAL simple undirected edges=[[0, 1], [0, 18], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13], [13, 14], [14, 15], [15, 16], [16, 17], [17, 18]]

edge indices are EXACT original list positions0..18;retain ALL vertices;no12-vertex or10-edge bound

original walk step=0;consecutive vertices=[0, 1];undirected candidate edge=[0, 1]

matching original edge index=0;recorded edge prefix=[0];length-and-adjacency status=是

original walk step=1;consecutive vertices=[1, 2];undirected candidate edge=[1, 2]

matching original edge index=2;recorded edge prefix=[0, 2];length-and-adjacency status=是

original walk step=2;consecutive vertices=[2, 3];undirected candidate edge=[2, 3]

matching original edge index=3;recorded edge prefix=[0, 2, 3];length-and-adjacency status=是

original walk step=3;consecutive vertices=[3, 4];undirected candidate edge=[3, 4]

matching original edge index=4;recorded edge prefix=[0, 2, 3, 4];length-and-adjacency status=是

original walk step=4;consecutive vertices=[4, 5];undirected candidate edge=[4, 5]

matching original edge index=5;recorded edge prefix=[0, 2, 3, 4, 5];length-and-adjacency status=是

original walk step=5;consecutive vertices=[5, 6];undirected candidate edge=[5, 6]

matching original edge index=6;recorded edge prefix=[0, 2, 3, 4, 5, 6];length-and-adjacency status=是

original walk step=6;consecutive vertices=[6, 7];undirected candidate edge=[6, 7]

matching original edge index=7;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7];length-and-adjacency status=是

original walk step=7;consecutive vertices=[7, 8];undirected candidate edge=[7, 8]

matching original edge index=8;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8];length-and-adjacency status=是

original walk step=8;consecutive vertices=[8, 9];undirected candidate edge=[8, 9]

matching original edge index=9;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9];length-and-adjacency status=是

original walk step=9;consecutive vertices=[9, 10];undirected candidate edge=[9, 10]

matching original edge index=10;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10];length-and-adjacency status=是

original walk step=10;consecutive vertices=[10, 11];undirected candidate edge=[10, 11]

matching original edge index=11;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];length-and-adjacency status=是

original walk step=11;consecutive vertices=[11, 12];undirected candidate edge=[11, 12]

matching original edge index=12;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];length-and-adjacency status=是

original walk step=12;consecutive vertices=[12, 13];undirected candidate edge=[12, 13]

matching original edge index=13;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];length-and-adjacency status=是

original walk step=13;consecutive vertices=[13, 14];undirected candidate edge=[13, 14]

matching original edge index=14;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];length-and-adjacency status=是

original walk step=14;consecutive vertices=[14, 15];undirected candidate edge=[14, 15]

matching original edge index=15;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];length-and-adjacency status=是

original walk step=15;consecutive vertices=[15, 16];undirected candidate edge=[15, 16]

matching original edge index=16;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];length-and-adjacency status=是

original walk step=16;consecutive vertices=[16, 17];undirected candidate edge=[16, 17]

matching original edge index=17;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];length-and-adjacency status=是

original walk step=17;consecutive vertices=[17, 18];undirected candidate edge=[17, 18]

matching original edge index=18;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];length-and-adjacency status=是

original walk step=18;consecutive vertices=[18, 0];undirected candidate edge=[0, 18]

matching original edge index=1;recorded edge prefix=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1];length-and-adjacency status=是

original complete walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0];required vertex-list length=20;actual length=20;found ORIGINAL edge indices=[0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1]

length-and-all-adjacency=是;recorded indices unique=是;recorded complete single-use coverage=是;vertex revisited=是;VALID Euler trail=是

逐步反向核對

• S022 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S021 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S020 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S019 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S018 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S017 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S016 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S015 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S014 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S013 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S012 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S011 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S010 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S009 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S008 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S007 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S006 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S005 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S004 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S003 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S002 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。
• S001 · premise_and_result_audit — 回到原始資料與本步條件核對，不把單向推論誤稱等價。

結論

([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1], 是, 是, 是, 是, 是, trail_allows_repeated_vertices_not_edges)

依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。

本題覆核與來源綁定

覆核日期：2026-10-26

原題、答案、步驟表達式、數值 AST 與既有歷史保持原樣；原來的候選狀態屬於來源歷史，不覆寫為形式化證明。