# Complete Walk Qualification, Simple Paths, Step Lengths and Repetition Boundaries

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

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

batch-20261014-advance-20261027-r07-graph-walk-simple-path-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.



## 1. Check Every Consecutive Step in a Vertex Sequence

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=[0, 1, 0, 1, 1]. 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

View answer and explanation([0, 1, 0, 1, 1], [[3, 1, 1]], No)

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 1, 0, 1, 1], [[3, 1, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0001/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 2. Check Every Consecutive Step in a Vertex Sequence

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=[2, 1, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 1, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0002/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 3. Check Every Consecutive Step in a Vertex Sequence

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=[3, 1, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 1, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0003/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 4. Check Every Consecutive Step in a Vertex Sequence

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=[4, 1, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 1, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0004/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 5. Check Every Consecutive Step in a Vertex Sequence

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, 1, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 1, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0005/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 6. Check Every Consecutive Step in a Vertex Sequence

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=[0, 2, 0, 1, 1]. 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 2, 0, 1, 1], [[3, 1, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0006/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 7. Check Every Consecutive Step in a Vertex Sequence

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=[1, 2, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 2, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0007/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 8. Check Every Consecutive Step in a Vertex Sequence

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=[3, 2, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 2, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0008/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 9. Check Every Consecutive Step in a Vertex Sequence

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=[4, 2, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 2, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0009/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 10. Check Every Consecutive Step in a Vertex Sequence

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, 2, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 2, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0010/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 11. Check Every Consecutive Step in a Vertex Sequence

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=[0, 3, 0, 1, 1]. 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

View answer and explanation([0, 3, 0, 1, 1], [[3, 1, 1]], No)

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 3, 0, 1, 1], [[3, 1, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0011/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 12. Check Every Consecutive Step in a Vertex Sequence

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=[1, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 3, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0012/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 13. Check Every Consecutive Step in a Vertex Sequence

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=[2, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 3, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0013/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 14. Check Every Consecutive Step in a Vertex Sequence

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=[4, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 3, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0014/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 15. Check Every Consecutive Step in a Vertex Sequence

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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 3, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0015/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 16. Check Every Consecutive Step in a Vertex Sequence

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=[0, 4, 0, 1, 1]. 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

View answer and explanation([0, 4, 0, 1, 1], [[3, 1, 1]], No)

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 4, 0, 1, 1], [[3, 1, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0016/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 17. Check Every Consecutive Step in a Vertex Sequence

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=[1, 4, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 4, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0017/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 18. Check Every Consecutive Step in a Vertex Sequence

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=[2, 4, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 4, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0018/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 19. Check Every Consecutive Step in a Vertex Sequence

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=[3, 4, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 4, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0019/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 20. Check Every Consecutive Step in a Vertex Sequence

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, 4, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([5, 4, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0020/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 21. Check Every Consecutive Step in a Vertex Sequence

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=[0, 5, 0, 1, 1]. 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

View answer and explanation([0, 5, 0, 1, 1], [[3, 1, 1]], No)

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([0, 5, 0, 1, 1], [[3, 1, 1]], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0021/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 22. Check Every Consecutive Step in a Vertex Sequence

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=[1, 5, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([1, 5, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0022/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 23. Check Every Consecutive Step in a Vertex Sequence

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=[2, 5, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([2, 5, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0023/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 24. Check Every Consecutive Step in a Vertex Sequence

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=[3, 5, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([3, 5, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0024/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 25. Check Every Consecutive Step in a Vertex Sequence

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=[4, 5, 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 (vertices,all bad [zero-based step index,from,to] records,walk?).

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

Look up every consecutive unordered pair in the complete edge set. An immediate repeated vertex is a bad step because loops are forbidden. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is ([4, 5, 0, 1, 0], [], Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0025/)

Graph walks and simple paths · qf-mod27-walk-qualification · candidate





## 26. A Walk Is Simple Only when Every Vertex Is Distinct

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=[0, 1, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0026/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 27. A Walk Is Simple Only when Every Vertex Is Distinct

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=[2, 1, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0027/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 28. A Walk Is Simple Only when Every Vertex Is Distinct

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=[3, 1, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0028/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 29. A Walk Is Simple Only when Every Vertex Is Distinct

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=[4, 1, 0, 1, 4]. 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?).

View answer and explanation(Yes, [1, 4], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [1, 4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0029/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 30. A Walk Is Simple Only when Every Vertex Is Distinct

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, 1, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0030/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 31. A Walk Is Simple Only when Every Vertex Is Distinct

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=[0, 2, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0031/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 32. A Walk Is Simple Only when Every Vertex Is Distinct

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=[1, 2, 0, 1, 1]. 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?).

View answer and explanation(No, [1], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (No, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0032/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 33. A Walk Is Simple Only when Every Vertex Is Distinct

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=[3, 2, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0033/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 34. A Walk Is Simple Only when Every Vertex Is Distinct

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=[4, 2, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0034/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 35. A Walk Is Simple Only when Every Vertex Is Distinct

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, 2, 0, 1, 5]. 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?).

View answer and explanation(Yes, [5], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [5], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0035/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 36. A Walk Is Simple Only when Every Vertex Is Distinct

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=[0, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0036/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 37. A Walk Is Simple Only when Every Vertex Is Distinct

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=[1, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0037/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 38. A Walk Is Simple Only when Every Vertex Is Distinct

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=[2, 3, 0, 1, 2]. 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?).

View answer and explanation(Yes, [2], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [2], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0038/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 39. A Walk Is Simple Only when Every Vertex Is Distinct

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=[4, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0039/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 40. A Walk Is Simple Only when Every Vertex Is Distinct

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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0040/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 41. A Walk Is Simple Only when Every Vertex Is Distinct

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=[0, 4, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0041/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 42. A Walk Is Simple Only when Every Vertex Is Distinct

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=[1, 4, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0, 1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0042/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 43. A Walk Is Simple Only when Every Vertex Is Distinct

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=[2, 4, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0043/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 44. A Walk Is Simple Only when Every Vertex Is Distinct

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=[3, 4, 0, 1, 3]. 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?).

View answer and explanation(Yes, [3], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [3], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0044/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 45. A Walk Is Simple Only when Every Vertex Is Distinct

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, 4, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0045/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 46. A Walk Is Simple Only when Every Vertex Is Distinct

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=[0, 5, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0046/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 47. A Walk Is Simple Only when Every Vertex Is Distinct

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=[1, 5, 0, 1, 1]. 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?).

View answer and explanation(No, [1], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (No, [1], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0047/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 48. A Walk Is Simple Only when Every Vertex Is Distinct

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=[2, 5, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0048/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 49. A Walk Is Simple Only when Every Vertex Is Distinct

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=[3, 5, 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?).

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

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [0], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0049/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 50. A Walk Is Simple Only when Every Vertex Is Distinct

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=[4, 5, 0, 1, 4]. 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?).

View answer and explanation(Yes, [4], No)

First check all adjacency obligations, then the no-repeated-vertex obligation. A list of distinct vertices with a missing edge is still not a simple path. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, [4], No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0050/)

Graph walks and simple paths · qf-mod27-walk-simple-path-qualification · candidate





## 51. Separate Candidate Length and Endpoints from Walk Qualification

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=[0, 1, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0051/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 52. Separate Candidate Length and Endpoints from Walk Qualification

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=[2, 1, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0052/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 53. Separate Candidate Length and Endpoints from Walk Qualification

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=[3, 1, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0053/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 54. Separate Candidate Length and Endpoints from Walk Qualification

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=[4, 1, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0054/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 55. Separate Candidate Length and Endpoints from Walk Qualification

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, 1, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0055/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 56. Separate Candidate Length and Endpoints from Walk Qualification

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=[0, 2, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0056/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 57. Separate Candidate Length and Endpoints from Walk Qualification

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=[1, 2, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0057/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 58. Separate Candidate Length and Endpoints from Walk Qualification

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=[3, 2, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0058/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 59. Separate Candidate Length and Endpoints from Walk Qualification

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=[4, 2, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0059/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 60. Separate Candidate Length and Endpoints from Walk Qualification

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, 2, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0060/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 61. Separate Candidate Length and Endpoints from Walk Qualification

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=[0, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0061/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 62. Separate Candidate Length and Endpoints from Walk Qualification

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=[1, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0062/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 63. Separate Candidate Length and Endpoints from Walk Qualification

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=[2, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0063/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 64. Separate Candidate Length and Endpoints from Walk Qualification

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=[4, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0064/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 65. Separate Candidate Length and Endpoints from Walk Qualification

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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0065/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 66. Separate Candidate Length and Endpoints from Walk Qualification

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=[0, 4, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0066/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 67. Separate Candidate Length and Endpoints from Walk Qualification

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=[1, 4, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0067/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 68. Separate Candidate Length and Endpoints from Walk Qualification

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=[2, 4, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0068/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 69. Separate Candidate Length and Endpoints from Walk Qualification

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=[3, 4, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0069/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 70. Separate Candidate Length and Endpoints from Walk Qualification

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, 4, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 5, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0070/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 71. Separate Candidate Length and Endpoints from Walk Qualification

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=[0, 5, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 0, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0071/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 72. Separate Candidate Length and Endpoints from Walk Qualification

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=[1, 5, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 1, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0072/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 73. Separate Candidate Length and Endpoints from Walk Qualification

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=[2, 5, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 2, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0073/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 74. Separate Candidate Length and Endpoints from Walk Qualification

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=[3, 5, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 3, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0074/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 75. Separate Candidate Length and Endpoints from Walk Qualification

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=[4, 5, 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?,edge_length,start,end,edge_length_equals_vertex_count_minus_one). Even for an invalid candidate retain the number of proposed steps and its two endpoints; its No flag means these do not certify a walk.

View answer and explanation(Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one)

Subtract one from the length of the nonempty vertex list. A single listed vertex has zero steps and identical endpoints. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, 4, 0, edge_length_equals_vertex_count_minus_one).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0075/)

Graph walks and simple paths · qf-mod27-walk-length-endpoints · candidate





## 76. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[0, 1, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, Yes)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0076/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 77. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[2, 1, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0077/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 78. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[3, 1, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0078/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 79. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[4, 1, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0079/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 80. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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, 1, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0080/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 81. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[0, 2, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

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

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 2], [0, 1]], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0081/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 82. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[1, 2, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0082/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 83. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[3, 2, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0083/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 84. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[4, 2, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0084/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 85. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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, 2, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0085/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 86. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[0, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

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

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 3], [0, 1]], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0086/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 87. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[1, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0087/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 88. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[2, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0088/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 89. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[4, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0089/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 90. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0090/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 91. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[0, 4, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 4], [0, 1]], No, No, Yes)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 4], [0, 1]], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0091/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 92. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[1, 4, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0092/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 93. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[2, 4, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0093/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 94. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[3, 4, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0094/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 95. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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, 4, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0095/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 96. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[0, 5, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 5], [0, 1]], No, No, Yes)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 5], [0, 1]], No, No, Yes).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0096/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 97. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[1, 5, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0, 1], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0, 1], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0097/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 98. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[2, 5, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0098/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 99. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[3, 5, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0099/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate





## 100. Distinguish Repeated Vertices, Repeated Edges and Closed Walks

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=[4, 5, 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?,edge_length,sorted distinct repeated vertex labels,distinct repeated unordered edges in first-repeat encounter order,trail?,simple_path?,closed?). A trail here is a qualified walk without repeated undirected edges. closed requires a qualified positive-length walk with equal endpoints; it is not a simple-cycle certificate.

View answer and explanation(Yes, 4, [0], [[0, 1]], No, No, No)

Normalize each step edge as an unordered pair before testing repetition. A closed trail can repeat its endpoint and therefore is not a simple path; a backtrack repeats an undirected edge. The length-zero singleton is a trail and simple path but is not closed under this positive-length convention. An exact finite or coordinate slice does not automatically establish a general infinite theorem. The exact result, retaining the requested order, is (Yes, 4, [0], [[0, 1]], No, No, No).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261027-r07-graph-walk-simple-path-0100/)

Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261027-r07-graph-walk-simple-path-foundations/validation-report.json)