第 18901 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-odd-degree-classification；資料={"n":2,"edges":[[0,1]]}。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-odd-degree-classification；資料={"n":2,"edges":[[0,1]]}。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-odd-degree-classification;original snapshot and mathematics retained

explicit ordered source output fields=[degrees, odd_vertices, edge_support_connected, euler_type, criterion]

complete ORIGINAL carrier Fin(2);canonical ORIGINAL simple undirected edges=[[0, 1]]

edge indices are EXACT original list positions0..0;retain ALL vertices;no12-vertex or10-edge bound

Euler vertex0 exact incident edge count := sum(1)

Euler vertex0 exact incident edge count = 1

original vertex=0;ALL original incident edge indices=[0]

degree=1;nonisolated=是;odd=是

Euler vertex1 exact incident edge count := sum(1)

Euler vertex1 exact incident edge count = 1

original vertex=1;ALL original incident edge indices=[0]

degree=1;nonisolated=是;odd=是

nonisolated edge support:original edge index=0;edge=[0, 1];from=0;new vertex=1

original-edge path from0=[0, 1];reached=[0, 1]

nonisolated edge support:complete permitted carrier=[0, 1];reached=[0, 1];unreached=[];ALL original edges=[[0, 1]]

no edge in carrier crosses reached/unreached cut;connected=是

whole original carrier:original edge index=0;edge=[0, 1];from=0;new vertex=1

original-edge path from0=[0, 1];reached=[0, 1]

whole original carrier:complete permitted carrier=[0, 1];reached=[0, 1];unreached=[];ALL original edges=[[0, 1]]

no edge in carrier crosses reached/unreached cut;connected=是

complete degree list=[1, 1];ALL odd vertices=[0, 1];actual nonisolated support=[0, 1];support connected=是;whole graph connected=是

ANY full-edge trail joins all nonisolated vertices in one support component;each internal visit pairs incident edges,leaving only0 or2 odd endpoints

Hierholzer stack top=0;first unused incident ORIGINAL edge index=0;edge=[0, 1]

mark edge used ONCE;push vertex=1;current stack=[0, 1];used edge set=[0]

Hierholzer current stack=[0, 1];top vertex has NO unused incident edge

finalize top into reverse output=[1];remaining stack=[0]

Hierholzer current stack=[0];top vertex has NO unused incident edge

finalize top into reverse output=[1, 0];remaining stack=[]

connected nonisolated support=[0, 1];odd vertices=[0, 1];finished reversed construction

ACTUAL Euler witness walk=[0, 1];actual ORIGINAL edge order=[0];every original edge once;endpoints have required parity

逐步反向核對

• 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 — 以精確有理數重算 Euler vertex1 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex0 exact incident edge count，不以小數近似替代。
• 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

([1, 1], [0, 1], 是, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

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。