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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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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 $\delta(G)\ge2$ , then $G$ contains a cycle.

Facts & Assumptions

Given: A finite simple graph $G$ with $\delta(G)\ge2$ .

[L1]

$G$ contains a path with at least $\delta(G)$ edges (Every nonempty finite simple graph contains a path with at least $\delta(G)$ 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 $P=(v_0,\ldots,v_\ell)$ , so $\ell\ge2$ .

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 $\deg_G(v_0)\ge2$ , besides $v_1$ there is a neighbour $v_j$ with $j\ge2$ .

step 1.1given

3.1

The list $(v_0,v_1,\ldots,v_j,v_0)$ is a closed walk of length $j+1\ge3$ 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 10 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)