TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

How statement and proof provenance work

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 nonempty finite simple graph contains a path with at least $\delta(G)$ edges

Statement

Every nonempty finite simple graph $G$ contains a path having at least $\delta(G)$ edges.

Facts & Assumptions

Given: A nonempty finite simple graph $G=(V,E)$ .

[F1]

$\delta(G)$ is the least degree of a vertex of $G$ (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[F2]

A path has no repeated vertex, and appending a neighbour not already on it produces a longer path (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

[F3]

A finite graph has only finitely many paths, so a path of greatest length exists (The cardinality $\lvert A\rvert$ of a finite set).

Proof

technique · constructive

1.1

Choose a path $P=(v_0,v_1,\ldots,v_\ell)$ of greatest length in $G$ .

givenF3chooseconstruct

2.1

Every neighbour of $v_0$ lies among $v_1,\ldots,v_\ell$ : a neighbour outside this list could be prepended to $P$ , giving a longer path and contradicting step 1.1.

step 1.1F2

3.1

The $\deg_G(v_0)$ distinct neighbours of $v_0$ therefore occupy distinct positions among the $\ell$ vertices $v_1,\ldots,v_\ell$ , so $\ell\ge\deg_G(v_0)\ge\delta(G)$ by [F1]. Thus $P$ has at least $\delta(G)$ edges.

step 2.1F1discharge-construct∎

Depends on

Used by

Every finite simple graph of minimum degree at least two contains a cycle Corollary

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

Cornell Mathematics, Graph Theory notes (standard reference, not scraped)

Every nonempty finite simple graph contains a path with at least δ(G)\delta(G)δ(G) edges