Complete Walk Qualification, Simple Paths, Step Lengths and Repetition Boundaries
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Graph walks and simple paths · qf-mod27-walk-repetition-classification · candidate
Canonical question JSONL · English JSONL · Validation report