Math 2.0

Combinatorics Trees, paths, cycles, connectivity Theorem

Dirac's theorem on Hamiltonian cycles

Builds on Minimum degreeHamiltonian graph

Statement

Let GG be a finite simple graph with n≥3n \ge 3 vertices. Suppose that every vertex of GG has degree at least n/2n/2. Then GG is Hamiltonian.

Remarks

The hypothesis says that the minimum degree is at least n/2n/2. The theorem follows from Ore's theorem: if all degrees are at least n/2n/2, then each sum of two degrees is at least nn. The bound is sharp. The complete bipartite graph Kk,k+1K_{k, k+1} has minimum degree k<(2k+1)/2k < (2k+1)/2, and it is not Hamiltonian.

History

G. A. Dirac published the theorem in 1952.

Reading path

To understand this page from the ground up, read these 11 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 levelTraildefinitionMinimum degree, definition, no evidence levelMinimum degreedefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionHamiltonian cycle, definition, no evidence levelHamiltonian cycledefinitionHamiltonian graph, definition, no evidence levelHamiltonian graphdefinitionDirac's theorem on Hamiltonian cycles, statement, no evidence levelDirac's theorem on Ha…statement
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. Minimum degree definition
  7. Circuit definition
  8. Cycle definition
  9. Hamiltonian cycle definition
  10. Hamiltonian graph definition
  11. Dirac's theorem on Hamiltonian cycles statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Hamiltonian_path, Bondy–Chvátal theorem
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