CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Every Hamiltonian graph on at least three vertices is $2$ -connected

Statement

Every Hamiltonian finite simple graph $G$ with $∣ V (G) ∣ \geq 3$ satisfies $κ (G) \geq 2$ ; equivalently, it is $2$ -connected.

Facts & Assumptions

Given: A Hamiltonian graph $G$ on at least three vertices.

[F1]

A Hamiltonian graph has a cycle containing all its vertices and is connected (Hamilton paths, Hamilton cycles, Hamiltonian graphs and Hamilton-connected graphs).

[L1]

Deleting any nonempty proper vertex set $S$ from a Hamiltonian graph leaves at most $∣ S ∣$ components (If $G$ has a Hamilton cycle, then deleting any nonempty proper vertex set $S$ leaves at most $∣ S ∣$ connected components).

[F2]

A vertex cut disconnects the graph or leaves at most one vertex, and $κ (G)$ is the minimum size of a vertex cut (Vertex cuts, edge cuts, vertex connectivity $κ (G)$ and edge connectivity $λ (G)$ , with conventions for complete and one-vertex graphs).

Proof

technique · direct

1.1

The graph $G$ is connected by [F1]. For any vertex $v$ , the set ${v}$ is nonempty and proper, and $G - v$ has at least two vertices. By [L1], it has at most one connected component, hence it is connected.

givenF1L1

2.1

Thus neither the empty set nor a singleton is a vertex cut: deleting nothing preserves connectedness, and deleting one vertex leaves a connected graph with at least two vertices.

step 1.1F2

3.1

Every vertex cut therefore has size at least two, so $κ (G) \geq 2$ by [F2].

step 2.1F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 13 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

Applied Combinatorics, Eulerian and Hamiltonian Graphs (standard reference, not scraped)

Every Hamiltonian graph on at least three vertices is 22-connected