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Question 13390: A Walk Is Simple Only when Every Vertex Is Distinct

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0025-13390 · q-20261014-advance-20261027-r07-graph-walk-simple-path-0040

Original question and givens

Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).

  • G001 — Use the complete finite undirected loop-free simple graph on Fin(6)={0,…,5} with edge set=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]] and nonempty ordered vertex sequence=[5, 3, 0, 1, 0]. A walk requires every consecutive pair to be adjacent. A simple path additionally forbids all repeated vertices. Edge length is the number of steps, not the number of listed vertices; a singleton vertex is a length-zero simple path. No directed-graph or general cycle theorem is claimed. Give (walk?,sorted distinct repeated vertex labels,simple_path?).

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.

Step-by-step derivation

S001 · implies

Before

carrier Fin(6);raw edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];original vertex sequence=[5, 3, 0, 1, 0]

After

canonical undirected simple edges=[[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [1, 2], [1, 3], [1, 4], [1, 5], [2, 3], [2, 4], [2, 5], [3, 4], [3, 5], [4, 5]];nonempty typed sequence retained

Conditions: none

Normalize undirected graph edges and reject loops/reversed duplicates.Preserve vertex-sequence order and repetitions.Repetition is not a parsing error in a candidate,though it may fail walk or simplicity qualification.

Step identifiers and dependencies

Rule: walk_simple_graph_and_ordered_candidate · Uses: G001

S002 · implies

Before

proposed step index=0;from=5;to=3;unordered step edge=[3, 5]

After

edge present=Yes

Conditions: none

Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.

Step identifiers and dependencies

Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S001

S003 · implies

Before

proposed step index=1;from=3;to=0;unordered step edge=[0, 3]

After

edge present=Yes

Conditions: none

Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.

Step identifiers and dependencies

Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S002

S004 · implies

Before

proposed step index=2;from=0;to=1;unordered step edge=[0, 1]

After

edge present=Yes

Conditions: none

Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.

Step identifiers and dependencies

Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S003

S005 · implies

Before

proposed step index=3;from=1;to=0;unordered step edge=[0, 1]

After

edge present=Yes

Conditions: none

Every consecutive pair must be adjacent in the original graph.An immediate repeated vertex proposes a diagonal step,which fails in a loop-free graph.Reversing direction does not change an undirected edge's identity.

Step identifiers and dependencies

Rule: walk_every_consecutive_pair_not_just_vertices · Uses: G001, S004

S006 · implies

Before

all bad proposed steps=[]

After

walk=Yes

Conditions: none

The whole step list must pass.A singleton has no proposed steps and passes vacuously as a valid length-zero walk.

Step identifiers and dependencies

Rule: complete_walk_candidate_qualification · Uses: G001, S005

S007 · implies

Before

original vertex sequence=[5, 3, 0, 1, 0]

After

sorted distinct repeated vertex labels=[0]

Conditions: none

A simple path forbids any repeated vertex,not only immediate repeats.This includes a closed walk's repeated endpoint.

Step identifiers and dependencies

Rule: simple_path_vertex_repetition_full_sequence · Uses: G001, S006

Reverse checks

  • 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

(Yes, [0], No)

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: 6a24f45563708b95a3592577120e56727f83b1e7c43fa9955e35a080bab0bede
  • candidate record SHA-256: 18ee8df3942994ef01e4b35d7ce8d9997adf6204045f2c9cabd1b67f3a5d1488
  • semantic review receipt SHA-256: 0e51c2bf621e6e998415fdd75d260464e625cecd5894a230c9f09553249551c5
  • question-first witness SHA-256: ceb79401a62782c9f7e5ed86afee9b1a71455f799bd24cf485cba7aa8ecaca8a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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