LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every walk between two vertices contains a path between the same endpoints

Statement

Let $W=(v_0,\ldots,v_\ell)$ be a walk in a graph. By deleting zero or more closed segments from its vertex list, one obtains a path from $v_0$ to $v_\ell$ of length at most $\ell$ . If $W$ repeats a vertex, the resulting path can be chosen to have length strictly less than $\ell$ .

Facts & Assumptions

Given: A walk $W=(v_0,\ldots,v_\ell)$ in a finite simple graph.

[F1]

A path is a walk with no repeated vertex (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

[F2]

If $v_i=v_j$ with $i<j$ , deleting $v_{i+1},\ldots,v_j$ leaves the list $(v_0,\ldots,v_i,v_{j+1},\ldots,v_\ell)$ , which is a walk with the same endpoints and length $\ell-(j-i)<\ell$ .

[F3]

A finite vertex list has only finitely many lists obtainable by deleting segments, so any nonempty family of their lengths has a least member (The cardinality $\lvert A\rvert$ of a finite set).

Proof

technique · constructive

1.1

Let $\mathcal W$ be the nonempty finite family of walks obtained from $W$ by deleting zero or more closed segments, and choose $P\in\mathcal W$ of least length.

givenF2F3chooseconstruct

2.1

The walk $P$ has no repeated vertex: if two positions carried the same vertex, [F2] would delete the intervening closed segment and produce a shorter member of $\mathcal W$ , contrary to the choice of $P$ .

step 1.1F2

3.1

By [F1], $P$ is a path with the same endpoints as $W$ , and its length is at most $\ell$ because $W\in\mathcal W$ . If $W$ repeats a vertex, applying [F2] once before minimizing gives a member of $\mathcal W$ shorter than $W$ , so $P$ is strictly shorter than $W$ .

step 1.1step 2.1F1F2discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

University of Lethbridge, Combinatorics: A Guided Tour, Paths and Cycles (standard reference, not scraped)