False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-07-31

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FALSE: vertex connectivity, edge connectivity and minimum degree are always equal

Statement

FALSE. Every nontrivial connected finite simple graph satisfies $κ (G) = λ (G) = δ (G)$ .

Facts & Assumptions

Given: The bowtie graph $G$ with two triangles $x a b x$ and $x c d x$ sharing only the vertex $x$ .

[L1]

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

[F1]

Vertex and edge connectivity are the minimum sizes of vertex and edge cuts (Vertex cuts, edge cuts, vertex connectivity $κ (G)$ and edge connectivity $λ (G)$ , with conventions for complete and one-vertex graphs).

Refutation

technique · direct

1.1

Deleting $x$ separates the two edges $a b$ and $c d$ , while the connected graph has no vertex cut of size $0$ . Hence $κ (G) = 1$ .

givenF1

1.2

Deleting the two edges $x a$ and $x b$ separates $a, b$ from the other triangle, so $λ (G) \leq 2$ . No single edge disconnects $G$ , because every edge lies on one of the two triangles and the other two edges of that triangle give an alternate path between its endpoints. Hence $λ (G) = 2$ .

givenF1

1.3

The four noncentral vertices have degree $2$ and $x$ has degree $4$ , so $δ (G) = 2$ .

given

2.1

Thus $κ (G) = 1 < 2 = λ (G) = δ (G)$ , a strict instance of the first Whitney inequality and a counterexample to the asserted equality.

step 1.1step 1.2step 1.3L1∎

Depends on

Used by

Two dense lobes meeting at one cut vertex give κ(G)=1<λ(G)=2<δ(G)=3 Counterexample

Dependency tree · two levels

8 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)