Math 2.0

Combinatorics Proof

Proof of Dirac's theorem on Hamiltonian cycles from Ore's theorem

Builds on GraphDegree of a vertexHamiltonian graphOre's theorem

Proof

We deduce the theorem from Ore's theorem.

  1. Let GG be a finite simple graph with n≥3n \ge 3 vertices, and suppose that deg⁡(w)≥n/2\deg(w) \ge n/2 for every vertex ww, where deg⁡\deg is the degree.
  2. Let uu and vv be two distinct vertices of GG that are not adjacent.
  3. By Step 1, deg⁡(u)+deg⁡(v)≥n/2+n/2=n\deg(u) + \deg(v) \ge n/2 + n/2 = n.
  4. So GG satisfies the hypothesis of Ore's theorem: it has n≥3n \ge 3 vertices, and every pair of distinct nonadjacent vertices has degree sum at least nn.
  5. By Ore's theorem, GG is Hamiltonian.

Idea

Dirac's condition bounds each degree from below by n/2n/2. Adding two such bounds gives Ore's condition on pairs of nonadjacent vertices. So Dirac's theorem is the special case of Ore's theorem where every vertex is "rich" on its own.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first. The path takes the proof Proof of Ore's theorem.

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Complete graph, example, no evidence levelComplete graphexampleGraph, definition, no evidence levelGraphdefinitionNeighbourhood, definition, no evidence levelNeighbourhooddefinitionWalk, definition, no evidence levelWalkdefinitionDegree of a vertex, definition, no evidence levelDegree of a vertexdefinitionPath, definition, no evidence levelPathdefinitionTrail, definition, no evidence levelTraildefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionHamiltonian cycle, definition, no evidence levelHamiltonian cycledefinitionHamiltonian graph, definition, no evidence levelHamiltonian graphdefinitionProof of Ore's theorem, proof, no evidence levelProof of Ore's theoremproofOre's theorem, statement, no evidence levelOre's theoremstatementProof of Dirac's theorem on Hamiltonian cycles from Ore's theorem, proof, no evidence levelProof of Dirac's theo…proof
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Complete graph example
  2. Graph definition
  3. Neighbourhood definition
  4. Walk definition
  5. Degree of a vertex definition
  6. Path definition
  7. Trail definition
  8. Circuit definition
  9. Cycle definition
  10. Hamiltonian cycle definition
  11. Hamiltonian graph definition
  12. Proof of Ore's theorem proof
  13. Ore's theorem statement
  14. Proof of Dirac's theorem on Hamiltonian cycles from Ore's theorem proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Ore's_theorem, Related results
About this page
TagX8WAMEEJ4D
URIm2://X8WAMEEJ4D
Kindproof
WritingW0 stub
MSC05C45
ProvesDirac's theorem on Hamiltonian cycles
UsesGraph, Degree of a vertex, Hamiltonian graph, Ore's theorem
Changed4 November 2026

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