# Single-Use Edges, Degree Parity, Support Connectivity and Zero Circuits

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261105-r04-eulerian-trail-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(3)={0,…,2}, with canonical sorted edges=[[0, 1], [0, 2], [1, 2]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 0], [0, 2, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 0], [0, 2, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0001/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 2. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(4)={0,…,3}, with canonical sorted edges=[[0, 1], [0, 3], [1, 2], [2, 3]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 0], [0, 2, 3, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 0], [0, 2, 3, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0002/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 3. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(5)={0,…,4}, with canonical sorted edges=[[0, 1], [0, 4], [1, 2], [2, 3], [3, 4]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 0], [0, 2, 3, 4, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 0], [0, 2, 3, 4, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0003/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 4. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(6)={0,…,5}, with canonical sorted edges=[[0, 1], [0, 5], [1, 2], [2, 3], [3, 4], [4, 5]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 0], [0, 2, 3, 4, 5, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 0], [0, 2, 3, 4, 5, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0004/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 5. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(7)={0,…,6}, with canonical sorted edges=[[0, 1], [0, 6], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 0], [0, 2, 3, 4, 5, 6, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 0], [0, 2, 3, 4, 5, 6, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0005/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 6. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(8)={0,…,7}, with canonical sorted edges=[[0, 1], [0, 7], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 2, 3, 4, 5, 6, 7, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 0], [0, 2, 3, 4, 5, 6, 7, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0006/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 7. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(9)={0,…,8}, with canonical sorted edges=[[0, 1], [0, 8], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 0], [0, 2, 3, 4, 5, 6, 7, 8, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 0], [0, 2, 3, 4, 5, 6, 7, 8, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0007/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 8. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(10)={0,…,9}, with canonical sorted edges=[[0, 1], [0, 9], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0008/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 9. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(11)={0,…,10}, with canonical sorted edges=[[0, 1], [0, 10], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0009/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 10. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(12)={0,…,11}, with canonical sorted edges=[[0, 1], [0, 11], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0010/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 11. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [0, 12], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0011/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 12. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(14)={0,…,13}, with canonical sorted edges=[[0, 1], [0, 13], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11], [11, 12], [12, 13]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0012/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 13. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(15)={0,…,14}, with canonical sorted edges=[[0, 1], [0, 14], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0013/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 14. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(16)={0,…,15}, with canonical sorted edges=[[0, 1], [0, 15], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0014/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 15. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(17)={0,…,16}, with canonical sorted edges=[[0, 1], [0, 16], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0015/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 16. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(18)={0,…,17}, with canonical sorted edges=[[0, 1], [0, 17], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0016/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 17. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(19)={0,…,18}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([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], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([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], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0017/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 18. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(20)={0,…,19}, with canonical sorted edges=[[0, 1], [0, 19], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0018/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 19. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(21)={0,…,20}, with canonical sorted edges=[[0, 1], [0, 20], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0019/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 20. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(22)={0,…,21}, with canonical sorted edges=[[0, 1], [0, 21], [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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0020/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 21. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted edges=[[0, 1], [0, 22], [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], [21, 22]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0021/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 22. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(24)={0,…,23}, with canonical sorted edges=[[0, 1], [0, 23], [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], [21, 22], [22, 23]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0022/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 23. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(25)={0,…,24}, with canonical sorted edges=[[0, 1], [0, 24], [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], [21, 22], [22, 23], [23, 24]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0023/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 24. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(26)={0,…,25}, with canonical sorted edges=[[0, 1], [0, 25], [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], [21, 22], [22, 23], [23, 24], [24, 25]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0024/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 25. Revisiting Vertices Is Allowed; Repeating an Edge Is Not

Use the complete finite simple undirected graph on Fin(27)={0,…,26}, with canonical sorted edges=[[0, 1], [0, 26], [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], [21, 22], [22, 23], [23, 24], [24, 25], [25, 26]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Supplied vertex_walk=[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0]. Give (vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges)

Record edge indices for steps found in the parent graph. Check walk length and all adjacencies separately from uniqueness and exact coverage of those recorded edge indices. The final validity requires both the length/adjacency and complete single-use edge checks; repeated vertices alone do not refute an Euler trail. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 0], [0, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 1], Yes, Yes, Yes, Yes, Yes, trail_allows_repeated_vertices_not_edges).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0025/)

Euler trails and circuits · qf-mod36-euler-trail-audit · candidate





## 26. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(2)={0,…,1}, with canonical sorted edges=[[0, 1]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 1], [0, 1], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 1], [0, 1], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0026/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 27. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(3)={0,…,2}, with canonical sorted edges=[[0, 1], [1, 2]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 1], [0, 2], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 1], [0, 2], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0027/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 28. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(4)={0,…,3}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 1], [0, 3], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 1], [0, 3], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0028/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 29. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(5)={0,…,4}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 1], [0, 4], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 1], [0, 4], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0029/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 30. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(6)={0,…,5}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 1], [0, 5], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 1], [0, 5], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0030/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 31. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(7)={0,…,6}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 1], [0, 6], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 1], [0, 6], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0031/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 32. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(8)={0,…,7}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 1], [0, 7], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 1], [0, 7], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0032/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 33. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(9)={0,…,8}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 1], [0, 8], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0033/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 34. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(10)={0,…,9}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 9], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 9], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0034/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 35. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(11)={0,…,10}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 10], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 10], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0035/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 36. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(12)={0,…,11}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 11], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 11], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0036/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 37. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 12], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 12], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0037/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 38. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(14)={0,…,13}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 13], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 13], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0038/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 39. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(15)={0,…,14}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 14], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 14], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0039/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 40. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(16)={0,…,15}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 15], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 15], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0040/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 41. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(17)={0,…,16}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 16], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 16], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0041/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 42. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(18)={0,…,17}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 17], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 17], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0042/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 43. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(19)={0,…,18}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 18], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 18], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0043/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 44. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(20)={0,…,19}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 19], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 19], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0044/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 45. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(21)={0,…,20}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 20], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 20], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0045/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 46. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(22)={0,…,21}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 21], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0046/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 47. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted 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], [21, 22]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 22], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 22], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0047/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 48. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(24)={0,…,23}, with canonical sorted 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], [21, 22], [22, 23]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 23], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 23], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0048/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 49. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(25)={0,…,24}, with canonical sorted 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], [21, 22], [22, 23], [23, 24]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 24], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 24], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0049/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 50. Degree Parity Needs Connected Nonisolated Support

Use the complete finite simple undirected graph on Fin(26)={0,…,25}, with canonical sorted 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], [21, 22], [22, 23], [23, 24], [24, 25]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (degrees, odd_vertices, edge_support_connected, euler_type, criterion).

View answer and explanation([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 25], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support)

For a nonzero-edge graph, compute every degree, all odd vertices and edge-support connectivity. Connected support with zero odd vertices gives a circuit, with two gives an open Euler trail; otherwise none. Parity alone does not connect separated edge components. The exact result, retaining the requested order, is ([1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1], [0, 25], Yes, open_euler_trail, zero_or_two_odd_vertices_on_connected_edge_support).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0050/)

Euler trails and circuits · qf-mod36-euler-odd-degree-classification · candidate





## 51. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(3)={0,…,2}, with canonical sorted edges=[[0, 1]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1], Yes, No, [0, 1], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1], Yes, No, [0, 1], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0051/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 52. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(4)={0,…,3}, with canonical sorted edges=[[0, 1], [1, 2]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2], Yes, No, [0, 2], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2], Yes, No, [0, 2], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0052/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 53. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(5)={0,…,4}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3], Yes, No, [0, 3], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3], Yes, No, [0, 3], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0053/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 54. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(6)={0,…,5}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4], Yes, No, [0, 4], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4], Yes, No, [0, 4], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0054/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 55. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(7)={0,…,6}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5], Yes, No, [0, 5], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5], Yes, No, [0, 5], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0055/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 56. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(8)={0,…,7}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6], Yes, No, [0, 6], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6], Yes, No, [0, 6], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0056/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 57. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(9)={0,…,8}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7], Yes, No, [0, 7], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7], Yes, No, [0, 7], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0057/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 58. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(10)={0,…,9}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, [0, 8], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8], Yes, No, [0, 8], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0058/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 59. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(11)={0,…,10}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, No, [0, 9], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], Yes, No, [0, 9], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0059/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 60. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(12)={0,…,11}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, No, [0, 10], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], Yes, No, [0, 10], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0060/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 61. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(13)={0,…,12}, with canonical sorted edges=[[0, 1], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, [0, 11], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11], Yes, No, [0, 11], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0061/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 62. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(14)={0,…,13}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, No, [0, 12], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], Yes, No, [0, 12], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0062/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 63. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(15)={0,…,14}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], Yes, No, [0, 13], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], Yes, No, [0, 13], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0063/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 64. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(16)={0,…,15}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Yes, No, [0, 14], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14], Yes, No, [0, 14], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0064/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 65. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(17)={0,…,16}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], Yes, No, [0, 15], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], Yes, No, [0, 15], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0065/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 66. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(18)={0,…,17}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, No, [0, 16], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], Yes, No, [0, 16], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0066/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 67. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(19)={0,…,18}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], Yes, No, [0, 17], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], Yes, No, [0, 17], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0067/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 68. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(20)={0,…,19}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, No, [0, 18], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18], Yes, No, [0, 18], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0068/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 69. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(21)={0,…,20}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], Yes, No, [0, 19], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], Yes, No, [0, 19], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0069/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 70. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(22)={0,…,21}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Yes, No, [0, 20], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], Yes, No, [0, 20], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0070/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 71. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(23)={0,…,22}, with canonical sorted 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]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], Yes, No, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], Yes, No, [0, 21], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0071/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 72. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(24)={0,…,23}, with canonical sorted 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], [21, 22]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, No, [0, 22], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22], Yes, No, [0, 22], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0072/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 73. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(25)={0,…,24}, with canonical sorted 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], [21, 22], [22, 23]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], Yes, No, [0, 23], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], Yes, No, [0, 23], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0073/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 74. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(26)={0,…,25}, with canonical sorted 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], [21, 22], [22, 23], [23, 24]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], Yes, No, [0, 24], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24], Yes, No, [0, 24], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0074/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 75. Isolated Vertices Do Not Disconnect the Edge Support

Use the complete finite simple undirected graph on Fin(27)={0,…,26}, with canonical sorted 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], [21, 22], [22, 23], [23, 24], [24, 25]] and zero-based edge indices in that order. The edge support contains exactly the nonisolated vertices. A trail may revisit vertices but may not repeat an edge; an Euler trail uses every edge once. Isolated vertices do not invalidate connected edge support, and no Hamiltonian conclusion follows. Give (nonisolated_vertices, edge_support_connected, whole_graph_connected, odd_vertices, euler_type, boundary).

View answer and explanation([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], Yes, No, [0, 25], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity)

Return the complete nonisolated support and its connectivity separately from whole-carrier connectivity. Degree parity is applied to the connected edge support, not used to demand that every isolated vertex lie in the same graph component. This family requires at least one edge. The exact result, retaining the requested order, is ([0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], Yes, No, [0, 25], open_euler_trail, isolated_vertices_do_not_break_edge_support_connectivity).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0075/)

Euler trails and circuits · qf-mod36-euler-support-connectivity · candidate





## 76. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(1)={0,…,0} and no edges, with n=1≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(1, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (1, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0076/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 77. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(2)={0,…,1} and no edges, with n=2≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(2, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (2, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0077/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 78. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(3)={0,…,2} and no edges, with n=3≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(3, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (3, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0078/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 79. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(4)={0,…,3} and no edges, with n=4≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(4, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (4, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0079/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 80. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(5)={0,…,4} and no edges, with n=5≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(5, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (5, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0080/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 81. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(6)={0,…,5} and no edges, with n=6≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(6, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (6, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0081/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 82. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(7)={0,…,6} and no edges, with n=7≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(7, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (7, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0082/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 83. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(8)={0,…,7} and no edges, with n=8≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(8, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (8, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0083/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 84. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(9)={0,…,8} and no edges, with n=9≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(9, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (9, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0084/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 85. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(10)={0,…,9} and no edges, with n=10≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(10, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (10, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0085/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 86. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(11)={0,…,10} and no edges, with n=11≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(11, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (11, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0086/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 87. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(12)={0,…,11} and no edges, with n=12≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(12, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (12, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0087/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 88. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(13)={0,…,12} and no edges, with n=13≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(13, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (13, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0088/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 89. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(14)={0,…,13} and no edges, with n=14≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(14, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (14, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0089/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 90. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(15)={0,…,14} and no edges, with n=15≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(15, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (15, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0090/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 91. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(16)={0,…,15} and no edges, with n=16≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(16, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (16, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0091/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 92. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(17)={0,…,16} and no edges, with n=17≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(17, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (17, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0092/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 93. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(18)={0,…,17} and no edges, with n=18≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(18, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (18, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0093/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 94. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(19)={0,…,18} and no edges, with n=19≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(19, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (19, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0094/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 95. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(20)={0,…,19} and no edges, with n=20≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(20, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (20, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0095/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 96. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(21)={0,…,20} and no edges, with n=21≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(21, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (21, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0096/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 97. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(22)={0,…,21} and no edges, with n=22≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(22, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (22, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0097/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 98. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(23)={0,…,22} and no edges, with n=23≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(23, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (23, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0098/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 99. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(24)={0,…,23} and no edges, with n=24≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(24, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (24, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0099/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate





## 100. A Nonempty Vertex Set Admits the Declared Length-Zero Circuit

Use vertex carrier Fin(25)={0,…,24} and no edges, with n=25≥1. Give (vertex_count, edge_count, length_zero_circuit, convention, boundary).

View answer and explanation(25, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope)

Adopt a walk consisting of a single chosen vertex as a length-zero circuit using every edge vacuously. The empty-vertex graph is explicitly out of scope, and no Hamiltonian claim follows. The exact result, retaining the requested order, is (25, 0, Yes, length_zero_circuit_on_nonempty_vertex_set, empty_vertex_graph_out_of_scope).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261105-r04-eulerian-trail-0100/)

Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r04-eulerian-trail-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261105-r04-eulerian-trail-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261105-r04-eulerian-trail-foundations/validation-report.json)