True or false if a graph is connected then the graph contains a Hamiltonian circuit 

This statement is **false**.

A graph being connected means there is a path between any two vertices, but it does not guarantee the existence of a Hamiltonian circuit. A Hamiltonian circuit is a cycle that visits each vertex exactly once and returns to the starting vertex. There are many connected graphs that do not have such a circuit.

### Example:
Consider a star graph with 5 vertices: one central vertex connected to four others. This graph is connected, but there is no way to visit each vertex exactly once and return to the starting point, hence no Hamiltonian circuit exists.

Would you like more details or have any other questions? Here are some related topics:

1. What are sufficient conditions for a graph to have a Hamiltonian circuit?
2. What is the difference between Hamiltonian and Eulerian circuits?
3. Can a disconnected graph have a Hamiltonian path?
4. How to determine if a given graph has a Hamiltonian circuit?
5. What are common algorithms to find Hamiltonian circuits in graphs?

**Tip:** If you need to determine whether a graph has a Hamiltonian circuit, try using Dirac's or Ore's theorem, which provide specific conditions under which a graph will have a Hamiltonian circuit.

True or false if a graph is connected then the graph contains a Hamiltonian circuit

Explore the relationship between connected graphs and Hamiltonian circuits. Learn why a connected graph does not necessarily contain a Hamiltonian circuit, with examples like the star graph. Understand key theorems such as Dirac's and Ore's, which provide sufficient conditions for a Hamiltonian circuit.

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