False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $\kappa(G)=\lambda(G)=\delta(G)$ .

Facts & Assumptions

Given: The bowtie graph $G$ with two triangles $xabx$ and $xcdx$ sharing only the vertex $x$ .

[L1]

Whitney's theorem guarantees only $\kappa(G)\le\lambda(G)\le\delta(G)$ (Whitney's inequalities: $\kappa(G)\le\lambda(G)\le\delta(G)$ 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 $\kappa(G)$ and edge connectivity $\lambda(G)$ , with conventions for complete and one-vertex graphs).

Refutation

technique · direct

1.1

Deleting $x$ separates the two edges $ab$ and $cd$ , while the connected graph has no vertex cut of size $0$ . Hence $\kappa(G)=1$ .

givenF1

1.2

Deleting the two edges $xa$ and $xb$ separates $a,b$ from the other triangle, so $\lambda(G)\le2$ . 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 $\lambda(G)=2$ .

givenF1

1.3

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

given

2.1

Thus $\kappa(G)=1<2=\lambda(G)=\delta(G)$ , 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 14 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)