CorollaryStatement: AI-adaptedProof: AI-generatedprecheck 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 · two levels

9 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

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

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