第 18908 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":9,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]]}。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":9,"edges":[[0,1],[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8]]}。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(9);canonical ORIGINAL simple undirected edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]]

edge indices are EXACT original list positions0..7;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, 1)

Euler vertex1 exact incident edge count = 2

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

degree=2;nonisolated=是;odd=否

Euler vertex2 exact incident edge count := sum(1, 1)

Euler vertex2 exact incident edge count = 2

original vertex=2;ALL original incident edge indices=[1, 2]

degree=2;nonisolated=是;odd=否

Euler vertex3 exact incident edge count := sum(1, 1)

Euler vertex3 exact incident edge count = 2

original vertex=3;ALL original incident edge indices=[2, 3]

degree=2;nonisolated=是;odd=否

Euler vertex4 exact incident edge count := sum(1, 1)

Euler vertex4 exact incident edge count = 2

original vertex=4;ALL original incident edge indices=[3, 4]

degree=2;nonisolated=是;odd=否

Euler vertex5 exact incident edge count := sum(1, 1)

Euler vertex5 exact incident edge count = 2

original vertex=5;ALL original incident edge indices=[4, 5]

degree=2;nonisolated=是;odd=否

Euler vertex6 exact incident edge count := sum(1, 1)

Euler vertex6 exact incident edge count = 2

original vertex=6;ALL original incident edge indices=[5, 6]

degree=2;nonisolated=是;odd=否

Euler vertex7 exact incident edge count := sum(1, 1)

Euler vertex7 exact incident edge count = 2

original vertex=7;ALL original incident edge indices=[6, 7]

degree=2;nonisolated=是;odd=否

Euler vertex8 exact incident edge count := sum(1)

Euler vertex8 exact incident edge count = 1

original vertex=8;ALL original incident edge indices=[7]

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:original edge index=1;edge=[1, 2];from=1;new vertex=2

original-edge path from0=[0, 1, 2];reached=[0, 1, 2]

nonisolated edge support:original edge index=2;edge=[2, 3];from=2;new vertex=3

original-edge path from0=[0, 1, 2, 3];reached=[0, 1, 2, 3]

nonisolated edge support:original edge index=3;edge=[3, 4];from=3;new vertex=4

original-edge path from0=[0, 1, 2, 3, 4];reached=[0, 1, 2, 3, 4]

nonisolated edge support:original edge index=4;edge=[4, 5];from=4;new vertex=5

original-edge path from0=[0, 1, 2, 3, 4, 5];reached=[0, 1, 2, 3, 4, 5]

nonisolated edge support:original edge index=5;edge=[5, 6];from=5;new vertex=6

original-edge path from0=[0, 1, 2, 3, 4, 5, 6];reached=[0, 1, 2, 3, 4, 5, 6]

nonisolated edge support:original edge index=6;edge=[6, 7];from=6;new vertex=7

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7];reached=[0, 1, 2, 3, 4, 5, 6, 7]

nonisolated edge support:original edge index=7;edge=[7, 8];from=7;new vertex=8

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8]

nonisolated edge support:complete permitted carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8];unreached=[];ALL original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]]

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:original edge index=1;edge=[1, 2];from=1;new vertex=2

original-edge path from0=[0, 1, 2];reached=[0, 1, 2]

whole original carrier:original edge index=2;edge=[2, 3];from=2;new vertex=3

original-edge path from0=[0, 1, 2, 3];reached=[0, 1, 2, 3]

whole original carrier:original edge index=3;edge=[3, 4];from=3;new vertex=4

original-edge path from0=[0, 1, 2, 3, 4];reached=[0, 1, 2, 3, 4]

whole original carrier:original edge index=4;edge=[4, 5];from=4;new vertex=5

original-edge path from0=[0, 1, 2, 3, 4, 5];reached=[0, 1, 2, 3, 4, 5]

whole original carrier:original edge index=5;edge=[5, 6];from=5;new vertex=6

original-edge path from0=[0, 1, 2, 3, 4, 5, 6];reached=[0, 1, 2, 3, 4, 5, 6]

whole original carrier:original edge index=6;edge=[6, 7];from=6;new vertex=7

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7];reached=[0, 1, 2, 3, 4, 5, 6, 7]

whole original carrier:original edge index=7;edge=[7, 8];from=7;new vertex=8

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8]

whole original carrier:complete permitted carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8];unreached=[];ALL original edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]]

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

complete degree list=[1, 2, 2, 2, 2, 2, 2, 2, 1];ALL odd vertices=[0, 8];actual nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8];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 stack top=1;first unused incident ORIGINAL edge index=1;edge=[1, 2]

mark edge used ONCE;push vertex=2;current stack=[0, 1, 2];used edge set=[0, 1]

Hierholzer stack top=2;first unused incident ORIGINAL edge index=2;edge=[2, 3]

mark edge used ONCE;push vertex=3;current stack=[0, 1, 2, 3];used edge set=[0, 1, 2]

Hierholzer stack top=3;first unused incident ORIGINAL edge index=3;edge=[3, 4]

mark edge used ONCE;push vertex=4;current stack=[0, 1, 2, 3, 4];used edge set=[0, 1, 2, 3]

Hierholzer stack top=4;first unused incident ORIGINAL edge index=4;edge=[4, 5]

mark edge used ONCE;push vertex=5;current stack=[0, 1, 2, 3, 4, 5];used edge set=[0, 1, 2, 3, 4]

Hierholzer stack top=5;first unused incident ORIGINAL edge index=5;edge=[5, 6]

mark edge used ONCE;push vertex=6;current stack=[0, 1, 2, 3, 4, 5, 6];used edge set=[0, 1, 2, 3, 4, 5]

Hierholzer stack top=6;first unused incident ORIGINAL edge index=6;edge=[6, 7]

mark edge used ONCE;push vertex=7;current stack=[0, 1, 2, 3, 4, 5, 6, 7];used edge set=[0, 1, 2, 3, 4, 5, 6]

Hierholzer stack top=7;first unused incident ORIGINAL edge index=7;edge=[7, 8]

mark edge used ONCE;push vertex=8;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8];used edge set=[0, 1, 2, 3, 4, 5, 6, 7]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8];top vertex has NO unused incident edge

finalize top into reverse output=[8];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7];top vertex has NO unused incident edge

finalize top into reverse output=[8, 7];remaining stack=[0, 1, 2, 3, 4, 5, 6]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6];top vertex has NO unused incident edge

finalize top into reverse output=[8, 7, 6];remaining stack=[0, 1, 2, 3, 4, 5]

Hierholzer current stack=[0, 1, 2, 3, 4, 5];top vertex has NO unused incident edge

finalize top into reverse output=[8, 7, 6, 5];remaining stack=[0, 1, 2, 3, 4]

Hierholzer current stack=[0, 1, 2, 3, 4];top vertex has NO unused incident edge

finalize top into reverse output=[8, 7, 6, 5, 4];remaining stack=[0, 1, 2, 3]

Hierholzer current stack=[0, 1, 2, 3];top vertex has NO unused incident edge

finalize top into reverse output=[8, 7, 6, 5, 4, 3];remaining stack=[0, 1, 2]

Hierholzer current stack=[0, 1, 2];top vertex has NO unused incident edge

finalize top into reverse output=[8, 7, 6, 5, 4, 3, 2];remaining stack=[0, 1]

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

finalize top into reverse output=[8, 7, 6, 5, 4, 3, 2, 1];remaining stack=[0]

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

finalize top into reverse output=[8, 7, 6, 5, 4, 3, 2, 1, 0];remaining stack=[]

connected nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8];odd vertices=[0, 8];finished reversed construction

ACTUAL Euler witness walk=[0, 1, 2, 3, 4, 5, 6, 7, 8];actual ORIGINAL edge order=[0, 1, 2, 3, 4, 5, 6, 7];every original edge once;endpoints have required parity

逐步反向核對

• S057 · 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。
• S056 · 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。
• S055 · 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。
• S054 · 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。
• S053 · 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。
• S052 · 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。
• S051 · 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。
• S050 · 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。
• S049 · 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。
• S048 · 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。
• S047 · 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。
• S046 · 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。
• S045 · 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。
• S044 · 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。
• S043 · 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。
• S042 · 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。
• S041 · 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。
• S040 · 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。
• S039 · 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。
• S038 · 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。
• S037 · 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。
• S036 · 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。
• S035 · 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。
• S034 · 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。
• S033 · 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。
• S032 · 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。
• S031 · 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。
• S030 · 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。
• S029 · 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。
• S028 · 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。
• S027 · 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。
• S026 · 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。
• S025 · 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。
• S024 · 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。
• S023 · 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。
• 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 — 以精確有理數重算 Euler vertex8 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex7 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex6 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex5 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex4 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex3 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex2 exact incident edge count，不以小數近似替代。
• 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, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], 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。