第 18921 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":22,"edges":[[0,1],[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],[18,19],[19,20],[20,21]]}。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":22,"edges":[[0,1],[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],[18,19],[19,20],[20,21]]}。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(22);canonical ORIGINAL simple undirected edges=[[0, 1], [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], [18, 19], [19, 20], [20, 21]]

edge indices are EXACT original list positions0..20;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, 1)

Euler vertex8 exact incident edge count = 2

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

degree=2;nonisolated=是;odd=否

Euler vertex9 exact incident edge count := sum(1, 1)

Euler vertex9 exact incident edge count = 2

original vertex=9;ALL original incident edge indices=[8, 9]

degree=2;nonisolated=是;odd=否

Euler vertex10 exact incident edge count := sum(1, 1)

Euler vertex10 exact incident edge count = 2

original vertex=10;ALL original incident edge indices=[9, 10]

degree=2;nonisolated=是;odd=否

Euler vertex11 exact incident edge count := sum(1, 1)

Euler vertex11 exact incident edge count = 2

original vertex=11;ALL original incident edge indices=[10, 11]

degree=2;nonisolated=是;odd=否

Euler vertex12 exact incident edge count := sum(1, 1)

Euler vertex12 exact incident edge count = 2

original vertex=12;ALL original incident edge indices=[11, 12]

degree=2;nonisolated=是;odd=否

Euler vertex13 exact incident edge count := sum(1, 1)

Euler vertex13 exact incident edge count = 2

original vertex=13;ALL original incident edge indices=[12, 13]

degree=2;nonisolated=是;odd=否

Euler vertex14 exact incident edge count := sum(1, 1)

Euler vertex14 exact incident edge count = 2

original vertex=14;ALL original incident edge indices=[13, 14]

degree=2;nonisolated=是;odd=否

Euler vertex15 exact incident edge count := sum(1, 1)

Euler vertex15 exact incident edge count = 2

original vertex=15;ALL original incident edge indices=[14, 15]

degree=2;nonisolated=是;odd=否

Euler vertex16 exact incident edge count := sum(1, 1)

Euler vertex16 exact incident edge count = 2

original vertex=16;ALL original incident edge indices=[15, 16]

degree=2;nonisolated=是;odd=否

Euler vertex17 exact incident edge count := sum(1, 1)

Euler vertex17 exact incident edge count = 2

original vertex=17;ALL original incident edge indices=[16, 17]

degree=2;nonisolated=是;odd=否

Euler vertex18 exact incident edge count := sum(1, 1)

Euler vertex18 exact incident edge count = 2

original vertex=18;ALL original incident edge indices=[17, 18]

degree=2;nonisolated=是;odd=否

Euler vertex19 exact incident edge count := sum(1, 1)

Euler vertex19 exact incident edge count = 2

original vertex=19;ALL original incident edge indices=[18, 19]

degree=2;nonisolated=是;odd=否

Euler vertex20 exact incident edge count := sum(1, 1)

Euler vertex20 exact incident edge count = 2

original vertex=20;ALL original incident edge indices=[19, 20]

degree=2;nonisolated=是;odd=否

Euler vertex21 exact incident edge count := sum(1)

Euler vertex21 exact incident edge count = 1

original vertex=21;ALL original incident edge indices=[20]

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:original edge index=8;edge=[8, 9];from=8;new vertex=9

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

nonisolated edge support:original edge index=9;edge=[9, 10];from=9;new vertex=10

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

nonisolated edge support:original edge index=10;edge=[10, 11];from=10;new vertex=11

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

nonisolated edge support:original edge index=11;edge=[11, 12];from=11;new vertex=12

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

nonisolated edge support:original edge index=12;edge=[12, 13];from=12;new vertex=13

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

nonisolated edge support:original edge index=13;edge=[13, 14];from=13;new vertex=14

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

nonisolated edge support:original edge index=14;edge=[14, 15];from=14;new vertex=15

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

nonisolated edge support:original edge index=15;edge=[15, 16];from=15;new vertex=16

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

nonisolated edge support:original edge index=16;edge=[16, 17];from=16;new vertex=17

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

nonisolated edge support:original edge index=17;edge=[17, 18];from=17;new vertex=18

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

nonisolated edge support:original edge index=18;edge=[18, 19];from=18;new vertex=19

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

nonisolated edge support:original edge index=19;edge=[19, 20];from=19;new vertex=20

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

nonisolated edge support:original edge index=20;edge=[20, 21];from=20;new vertex=21

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]

nonisolated edge support:complete permitted carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];unreached=[];ALL original edges=[[0, 1], [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], [18, 19], [19, 20], [20, 21]]

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:original edge index=8;edge=[8, 9];from=8;new vertex=9

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

whole original carrier:original edge index=9;edge=[9, 10];from=9;new vertex=10

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

whole original carrier:original edge index=10;edge=[10, 11];from=10;new vertex=11

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

whole original carrier:original edge index=11;edge=[11, 12];from=11;new vertex=12

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

whole original carrier:original edge index=12;edge=[12, 13];from=12;new vertex=13

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

whole original carrier:original edge index=13;edge=[13, 14];from=13;new vertex=14

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

whole original carrier:original edge index=14;edge=[14, 15];from=14;new vertex=15

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

whole original carrier:original edge index=15;edge=[15, 16];from=15;new vertex=16

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

whole original carrier:original edge index=16;edge=[16, 17];from=16;new vertex=17

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

whole original carrier:original edge index=17;edge=[17, 18];from=17;new vertex=18

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

whole original carrier:original edge index=18;edge=[18, 19];from=18;new vertex=19

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

whole original carrier:original edge index=19;edge=[19, 20];from=19;new vertex=20

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

whole original carrier:original edge index=20;edge=[20, 21];from=20;new vertex=21

original-edge path from0=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]

whole original carrier:complete permitted carrier=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];reached=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];unreached=[];ALL original edges=[[0, 1], [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], [18, 19], [19, 20], [20, 21]]

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

complete degree list=[1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1];ALL odd vertices=[0, 21];actual nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];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 stack top=8;first unused incident ORIGINAL edge index=8;edge=[8, 9]

mark edge used ONCE;push vertex=9;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8]

Hierholzer stack top=9;first unused incident ORIGINAL edge index=9;edge=[9, 10]

mark edge used ONCE;push vertex=10;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Hierholzer stack top=10;first unused incident ORIGINAL edge index=10;edge=[10, 11]

mark edge used ONCE;push vertex=11;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Hierholzer stack top=11;first unused incident ORIGINAL edge index=11;edge=[11, 12]

mark edge used ONCE;push vertex=12;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Hierholzer stack top=12;first unused incident ORIGINAL edge index=12;edge=[12, 13]

mark edge used ONCE;push vertex=13;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Hierholzer stack top=13;first unused incident ORIGINAL edge index=13;edge=[13, 14]

mark edge used ONCE;push vertex=14;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

Hierholzer stack top=14;first unused incident ORIGINAL edge index=14;edge=[14, 15]

mark edge used ONCE;push vertex=15;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Hierholzer stack top=15;first unused incident ORIGINAL edge index=15;edge=[15, 16]

mark edge used ONCE;push vertex=16;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Hierholzer stack top=16;first unused incident ORIGINAL edge index=16;edge=[16, 17]

mark edge used ONCE;push vertex=17;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

Hierholzer stack top=17;first unused incident ORIGINAL edge index=17;edge=[17, 18]

mark edge used ONCE;push vertex=18;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

Hierholzer stack top=18;first unused incident ORIGINAL edge index=18;edge=[18, 19]

mark edge used ONCE;push vertex=19;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

Hierholzer stack top=19;first unused incident ORIGINAL edge index=19;edge=[19, 20]

mark edge used ONCE;push vertex=20;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Hierholzer stack top=20;first unused incident ORIGINAL edge index=20;edge=[20, 21]

mark edge used ONCE;push vertex=21;current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];used edge set=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];top vertex has NO unused incident edge

finalize top into reverse output=[21];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

Hierholzer current stack=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9];top vertex has NO unused incident edge

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9];remaining stack=[0, 1, 2, 3, 4, 5, 6, 7, 8]

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

finalize top into reverse output=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 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=[21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0];remaining stack=[]

connected nonisolated support=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];odd vertices=[0, 21];finished reversed construction

ACTUAL Euler witness walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21];actual ORIGINAL edge order=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20];every original edge once;endpoints have required parity

逐步反向核對

• S135 · 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。
• S134 · 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。
• S133 · 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。
• S132 · 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。
• S131 · 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。
• S130 · 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。
• S129 · 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。
• S128 · 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。
• S127 · 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。
• S126 · 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。
• S125 · 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。
• S124 · 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。
• S123 · 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。
• S122 · 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。
• S121 · 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。
• S120 · 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。
• S119 · 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。
• S118 · 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。
• S117 · 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。
• S116 · 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。
• S115 · 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。
• S114 · 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。
• S113 · 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。
• S112 · 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。
• S111 · 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。
• S110 · 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。
• S109 · 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。
• S108 · 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。
• S107 · 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。
• S106 · 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。
• S105 · 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。
• S104 · 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。
• S103 · 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。
• S102 · 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。
• S101 · 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。
• S100 · 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。
• S099 · 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。
• S098 · 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。
• S097 · 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。
• S096 · 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。
• S095 · 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。
• S094 · 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。
• S093 · 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。
• S092 · 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。
• S091 · 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。
• S090 · 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。
• S089 · 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。
• S088 · 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。
• S087 · 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。
• S086 · 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。
• S085 · 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。
• S084 · 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。
• S083 · 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。
• S082 · 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。
• S081 · 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。
• S080 · 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。
• S079 · 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。
• S078 · 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。
• S077 · 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。
• S076 · 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。
• S075 · 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。
• S074 · 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。
• S073 · 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。
• S072 · 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。
• S071 · 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。
• S070 · 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。
• S069 · 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。
• S068 · 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。
• S067 · 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。
• S066 · 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。
• S065 · 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。
• S064 · 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。
• S063 · 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。
• S062 · 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。
• S061 · 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。
• S060 · 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。
• S059 · 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。
• S058 · 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。
• 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 — 以精確有理數重算 Euler vertex21 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex20 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex19 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex18 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex17 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex16 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex15 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex14 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex13 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex12 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex11 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex10 exact incident edge count，不以小數近似替代。
• 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 — 以精確有理數重算 Euler vertex9 exact incident edge count，不以小數近似替代。
• 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], 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。