Math 2.0

Combinatorics Trees, paths, cycles, connectivity Theorem

Ore's theorem

Builds on Degree of a vertexHamiltonian graph

Statement

Let GG be a finite simple graph with n≥3n \ge 3 vertices. Suppose that deg⁡(u)+deg⁡(v)≥n\deg(u) + \deg(v) \ge n for any two distinct vertices uu and vv that are not adjacent. Then GG is Hamiltonian.

Remarks

Dirac's theorem is a special case. The Bondy–Chvátal theorem of 1976 extends it. It says that GG is Hamiltonian if and only if its closure is Hamiltonian. The closure comes from GG when you add edges uvuv with deg⁡(u)+deg⁡(v)≥n\deg(u) + \deg(v) \ge n again and again.

History

Øystein Ore proved the theorem in 1960.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionNeighbourhood, definition, no evidence levelNeighbourhooddefinitionWalk, definition, no evidence levelWalkdefinitionDegree of a vertex, definition, no evidence levelDegree of a vertexdefinitionTrail, definition, no evidence levelTraildefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionHamiltonian cycle, definition, no evidence levelHamiltonian cycledefinitionHamiltonian graph, definition, no evidence levelHamiltonian graphdefinitionOre's theorem, statement, no evidence levelOre's theoremstatement
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.
  • no evidence level
  1. Graph definition
  2. Neighbourhood definition
  3. Walk definition
  4. Degree of a vertex definition
  5. Trail definition
  6. Circuit definition
  7. Cycle definition
  8. Hamiltonian cycle definition
  9. Hamiltonian graph definition
  10. Ore's theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Ore's_theorem, Formal statement
About this page
Tag0NACHJYPKW
URIm2://0NACHJYPKW
Kindstatement, theorem
WritingW1 stated
MSC05C45
StatusThis copy of the site has no build data. Run m2 status 0NACHJYPKW.
UsesDegree of a vertex, Hamiltonian graph
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