# Question 18955: Revisiting Vertices Is Allowed; Repeating an Edge Is Not [Return to catalogue page 95](https://ocme.evemisslab.com/en/questions/page/95/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261105-r04-eulerian-trail-foundations/#q-20261014-advance-20261105-r04-eulerian-trail-0005) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0036-18955 · q-20261014-advance-20261105-r04-eulerian-trail-0005 ## Original question and givens 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). - G001 — 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). ## Definition and scope Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate. - [mko-eulerian-trail](https://ocme.evemisslab.com/en/mko/mko-eulerian-trail/) ## Step-by-step derivation S001 · impliesBefore source operation=euler-trail-audit;original snapshot and mathematics retained After explicit ordered source output fields=[vertex_walk, used_edge_indices, length_and_adjacency, recorded_edge_indices_unique, recorded_edges_cover_exactly_once, vertex_revisited, valid_euler_trail, boundary] Conditions: none Keep the historical interface marker in the original snapshot while listing the original ordered fields here.No JavaScript or external table is needed for the main text,and field labels are not mathematical evidence. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · impliesBefore complete ORIGINAL carrier Fin(7);canonical ORIGINAL simple undirected edges=[[0, 1], [0, 6], [1, 2], [2, 3], [3, 4], [4, 5], [5, 6]] After edge indices are EXACT original list positions0..6;retain ALL vertices;no12-vertex or10-edge bound Conditions: none Keep edge indices at their original numeric positions,not string-sorted vertex pairs.This family permits up to60 vertices rather than importing earlier bounds;empty-edge graphs appear only where the operation allows them. Step identifiers and dependenciesRule: euler_original_simple_graph_indices_full_carrier · Uses: G001, S001 S003 · impliesBefore original walk step=0;consecutive vertices=[0, 1];undirected candidate edge=[0, 1] After matching original edge index=0;recorded edge prefix=[0];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S002 S004 · impliesBefore original walk step=1;consecutive vertices=[1, 2];undirected candidate edge=[1, 2] After matching original edge index=2;recorded edge prefix=[0, 2];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S003 S005 · impliesBefore original walk step=2;consecutive vertices=[2, 3];undirected candidate edge=[2, 3] After matching original edge index=3;recorded edge prefix=[0, 2, 3];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S004 S006 · impliesBefore original walk step=3;consecutive vertices=[3, 4];undirected candidate edge=[3, 4] After matching original edge index=4;recorded edge prefix=[0, 2, 3, 4];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S005 S007 · impliesBefore original walk step=4;consecutive vertices=[4, 5];undirected candidate edge=[4, 5] After matching original edge index=5;recorded edge prefix=[0, 2, 3, 4, 5];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S006 S008 · impliesBefore original walk step=5;consecutive vertices=[5, 6];undirected candidate edge=[5, 6] After matching original edge index=6;recorded edge prefix=[0, 2, 3, 4, 5, 6];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S007 S009 · impliesBefore original walk step=6;consecutive vertices=[6, 0];undirected candidate edge=[0, 6] After matching original edge index=1;recorded edge prefix=[0, 2, 3, 4, 5, 6, 1];length-and-adjacency status=Yes Conditions: none Look up every consecutive step in the original graph.A missing edge fails adjacency and is not added as a fake used index.The recorded list contains only found edges,so its coverage cannot replace complete walk validity. Step identifiers and dependenciesRule: euler_every_walk_step_missing_edge_not_recorded · Uses: G001, S008 S010 · impliesBefore original complete walk=[0, 1, 2, 3, 4, 5, 6, 0];required vertex-list length=8;actual length=8;found ORIGINAL edge indices=[0, 2, 3, 4, 5, 6, 1] After length-and-all-adjacency=Yes;recorded indices unique=Yes;recorded complete single-use coverage=Yes;vertex revisited=Yes;VALID Euler trail=Yes Conditions: none An Euler trail may revisit vertices but not edges.Final validity conjoins length/all-adjacency with complete single-use edge coverage.Extra missing steps can leave a recorded list that covers every edge while invalidating the walk.One bad supplied walk does not disprove existence of another Euler walk. Step identifiers and dependenciesRule: euler_trail_full_conjunction_repeated_vertices_permitted · Uses: G001, S009 ## Reverse checks - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S005 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([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) The full substitutions and premise checks give the original answer; source identity,format and status are unchanged. ## This question review and source binding Review date: 2026-10-26 Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: db06579696ecb09bd04efbf20137584e155272e5b549aa7743edbd64673ddba0 - candidate record SHA-256: d3e285786e4f7addb91c6e4f9e3b351d2da29fc929750317dd48aa2da85ce858 - semantic review receipt SHA-256: efee5462991619d6ed9f64f11ddb085e5614ef33f9f8fcd8cf29b9329f8fd301 - question-first witness SHA-256: acf91fa1622f01d2cf04b2c513294254323665351f660f2628aaafdb869c4ff2 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261105-r04-eulerian-trail-0005.json)