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

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Whitney's inequalities: $\kappa(G)\le\lambda(G)\le\delta(G)$ for every nontrivial connected graph

Statement

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

$\kappa(G)\le\lambda(G)\le\delta(G).$

This includes complete graphs under the convention $\kappa(K_n)=n-1$ : for $K_n$ with $n\ge2$ , all three quantities equal $n-1$ .

Facts & Assumptions

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

[L1]

$\kappa(G)\le\lambda(G)$ for every nontrivial connected graph (For every nontrivial connected graph, $\kappa(G)\le\lambda(G)$ ).

[L2]

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

Proof

technique · direct

1.1

Applying [L1] and [L2] to $G$ gives $\kappa(G)\le\lambda(G)\le\delta(G)$ .

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 $\kappa(K_n)=n-1$ ; 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 $\lambda(K_n)=\delta(K_n)=n-1$ , as stated.

L1L2algebra∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 9 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)

Whitney's inequalities: κ(G)≤λ(G)≤δ(G)\kappa(G)\le\lambda(G)\le\delta(G)κ(G)≤λ(G)≤δ(G) for every nontrivial connected graph