TheoremStatement: Literature-sourcedProof: AI-generatedprecheck 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.

Whitney's inequalities: $κ (G) \leq λ (G) \leq δ (G)$ for every nontrivial connected graph

Statement

For every connected finite simple graph $G$ with at least two vertices,

$κ (G) \leq λ (G) \leq δ (G) .$

This includes complete graphs under the convention $κ (K_{n}) = n - 1$ : for $K_{n}$ with $n \geq 2$ , all three quantities equal $n - 1$ .

Facts & Assumptions

Given: A connected finite simple graph $G$ with at least two vertices.

[L1]

$κ (G) \leq λ (G)$ for every nontrivial connected graph (For every nontrivial connected graph, $κ (G) \leq λ (G)$ ).

[L2]

$λ (G) \leq δ (G)$ for every nontrivial connected graph (For every nontrivial connected graph, $λ (G) \leq δ (G)$ ).

Proof

technique · direct

1.1

Applying [L1] and [L2] to $G$ gives $κ (G) \leq λ (G) \leq δ (G)$ .

L1L2

2.1

For $K_{n}$ , deleting fewer than $n - 1$ vertices leaves a nonempty complete graph and deleting $n - 1$ leaves one vertex, so $κ (K_{n}) = n - 1$ ; every vertex has degree $n - 1$ , and deleting all $n - 1$ edges incident with one vertex is an edge cut, while any smaller edge deletion leaves every pair joined through a remaining direct edge or a two-edge path. Hence $λ (K_{n}) = δ (K_{n}) = n - 1$ , as stated.

L1L2algebra∎

Depends on

Used by

Dependency tree · two levels

7 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

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

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