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

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Dirac's theorem: every $n$ -vertex graph with $n\ge3$ and $\delta(G)\ge n/2$ is Hamiltonian

Statement

Every finite simple graph $G$ on $n\ge3$ vertices with $\delta(G)\ge n/2$ is Hamiltonian.

Facts & Assumptions

Given: An $n$ -vertex graph $G$ with $n\ge3$ and $\delta(G)\ge n/2$ .

[L1]

Dirac's minimum-degree hypothesis implies Ore's degree-sum condition (The minimum-degree condition $\delta(G)\ge n/2$ implies Ore's degree-sum condition).

[L2]

Ore's degree-sum condition on an $n$ -vertex graph with $n\ge3$ implies Hamiltonicity (Ore's theorem: an $n$ -vertex graph with $n\ge3$ and $\deg(u)+\deg(v)\ge n$ for every nonadjacent pair is Hamiltonian).

Proof

technique · direct

1.1

By [L1], every nonadjacent pair $u,v$ satisfies $\deg(u)+\deg(v)\ge n$ .

givenL1

2.1

Since $n\ge3$ , [L2] applies and shows that $G$ is Hamiltonian.

step 1.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 21 results over 15 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)

Dirac's theorem: every nnn-vertex graph with n≥3n\ge3n≥3 and δ(G)≥n/2\delta(G)\ge n/2δ(G)≥n/2 is Hamiltonian