Single-Use Edges, Degree Parity, Support Connectivity and Zero Circuits
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Euler trails and circuits · qf-mod36-euler-zero-edge-convention · candidate
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