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2. Hamiltonian Circuit
The lecture discusses Hamiltonian circuits and paths, emphasizing their importance in graph theory. It introduces Dirac's and Ore's theorems as sufficient conditions for the existence of Hamiltonian circuits within graphs, highlighting the differences compared to Eulerian graphs. A thorough explanation of both sufficient conditions is provided, alongside proofs to enhance understanding of their application and limitations.
Sections
A Hamiltonian circuit is a path in a graph that visits each vertex exactly once, returning to the starting vertex, while a Hamiltonian path visits each vertex exactly once without needing to return.
Hamiltonian circuits require each vertex of a graph to be visited exactly once without repetition.
Dirac's theorem states that a connected graph with each vertex degree at least n/2 is Hamiltonian.
Ore's condition relates to non-adjacent vertices and indicates that if the sum of their degrees is at least n, then the graph is Hamiltonian.
Hamiltonian Circuit
A simple circuit in a graph that visits every vertex exactly once and returns to the starting vertex.
Hamiltonian Path
A path in a graph that visits every vertex exactly once but does not necessarily return to the starting vertex.
Dirac's Theorem
States that a connected graph with each vertex having a degree of at least n/2 contains a Hamiltonian circuit.
Ore's Condition
States that for any non-adjacent vertices u and v in a graph, if the sum of their degrees is at least n, the graph is Hamiltonian.
Practice Exercises
Total Questions
2
Estimated Time
4 min
Passing Score
70%
Instructions
- Read each question carefully
- You can use hints if you need help
- Complete all questions before submitting
1 more question available
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