LemmaStatement: Literature-sourcedProof: AI-generatedSession-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.

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For every nontrivial connected graph, $\lambda(G)\le\delta(G)$

Statement

For every connected finite simple graph $G$ with at least two vertices, $\lambda(G)\le\delta(G)$ .

Facts & Assumptions

Given: A connected graph $G=(V,E)$ with $|V|\ge2$ .

[F1]

$\lambda(G)$ is the minimum size of an edge set whose deletion disconnects $G$ (Vertex cuts, edge cuts, vertex connectivity $\kappa(G)$ and edge connectivity $\lambda(G)$ , with conventions for complete and one-vertex graphs).

[F2]

A vertex $v$ of minimum degree has exactly $\delta(G)$ incident edges (Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

Proof

technique · direct

1.1

Choose a vertex $v$ with $\deg_G(v)=\delta(G)$ and let $F$ be the set of all edges incident with $v$ . Then $|F|=\delta(G)$ .

givenF2choose

2.1

In $G-F$ , the vertex $v$ is isolated while at least one other vertex remains, so $G-F$ is disconnected. Thus $F$ is an edge cut.

step 1.1F1

3.1

Minimality in [F1] gives $\lambda(G)\le|F|=\delta(G)$ .

step 1.1step 2.1F1∎

Depends on

Used by

Whitney's inequalities: κ(G)≤λ(G)≤δ(G) for every nontrivial connected graph Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 19 results over 12 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

Wolfram MathWorld, Vertex Connectivity (standard reference, not scraped)

For every nontrivial connected graph, λ(G)≤δ(G)\lambda(G)\le\delta(G)λ(G)≤δ(G)