CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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.

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

Statement

If $G$ is a finite simple graph with $δ (G) \geq 2$ , then $G$ contains a cycle.

Facts & Assumptions

Given: A finite simple graph $G$ with $δ (G) \geq 2$ .

[L1]

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

[F1]

A closed walk of length at least $3$ whose vertices before the final repeated endpoint are distinct is a cycle (Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges).

Proof

technique · direct

1.1

By [L1], $G$ has a path of length at least $2$ . Among all paths choose one of greatest length, say $P = (v_{0}, \dots, v_{ℓ})$ , so $ℓ \geq 2$ .

givenL1choose

2.1

Every neighbour of $v_{0}$ lies on $P$ , since a neighbour outside $P$ could be prepended to make a longer path. Since $\deg_{G} (v_{0}) \geq 2$ , besides $v_{1}$ there is a neighbour $v_{j}$ with $j \geq 2$ .

step 1.1given

3.1

The list $(v_{0}, v_{1}, \dots, v_{j}, v_{0})$ is a closed walk of length $j + 1 \geq 3$ with distinct vertices before its final $v_{0}$ , so it is a cycle by [F1].

step 1.1step 2.1F1∎

Depends on

Used by

Every nonempty forest has a vertex of degree at most one Lemma

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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