Hamiltonian and Eulerian Paths/Circuits

Instructions: Complete the following problems, showing all your work. Write your answer on a long-sized bondpaper.

Hamiltonian Paths and Circuits

A Hamiltonian path is a path that visits every vertex exactly once. A Hamiltonian circuit is a Hamiltonian path that starts and ends at the same vertex.

1. Given the graph below, determine if a Hamiltonian path exists. If it does, find one.

vertex

edge

2. For the same graph in problem 1, determine if a Hamiltonian circuit exists. If it does, find one.

3. Consider the complete graph K5 (a graph with 5 vertices where every pair of vertices is connected by an edge). Prove that K5 has a Hamiltonian circuit.

Eulerian Paths and Circuits

An Eulerian path is a path that traverses every edge exactly once. An Eulerian circuit is an Eulerian path that starts and ends at the same vertex.

4. Given the graph below, determine if an Eulerian path exists. If it does, find one.

Let's break down each problem related to Hamiltonian and Eulerian paths and circuits. Since I don't have access to specific graphs that you're referring to, I'll describe the general approach to solving these problems, which you can apply to the graphs you're working with.

### Hamiltonian Paths and Circuits

#### Problem 1: Determine if a Hamiltonian Path Exists
To check if a Hamiltonian path exists in the given graph, follow these steps:

1. **Definition Recap**: A Hamiltonian path is a path that visits every vertex exactly once.
2. **Step-by-step Process**:
   - Look at the structure of the graph and the number of vertices.
   - Determine if there’s a sequence of vertices you can traverse such that every vertex is visited exactly once.
   - If such a sequence exists, write it down.
   - Otherwise, explain why it is not possible (e.g., due to disconnected vertices, etc.).
   
   **Tip**: Hamiltonian paths do not have a simple necessary-and-sufficient condition, unlike Eulerian paths. You need to experiment with different paths or apply specific theorems, like Dirac's or Ore's theorem, which help in identifying if the graph has a Hamiltonian path.

#### Problem 2: Determine if a Hamiltonian Circuit Exists
For this part, you need to determine if the graph contains a Hamiltonian circuit. A Hamiltonian circuit is a Hamiltonian path that starts and ends at the same vertex.

1. **Step-by-step Process**:
   -

Hamiltonian and Eulerian Paths/Circuits

Instructions: Complete the following problems, showing all your work. Write your answer on a long-sized bondpaper.

Hamiltonian Paths and Circuits

A Hamiltonian path is a path that visits every vertex exactly once. A Hamiltonian circuit is a Hamiltonian path that starts and ends at the same vertex.

1. Given the graph below, determine if a Hamiltonian path exists. If it does, find one.

2. For the same graph in problem 1, determine if a Hamiltonian circuit exists. If it does, find one.

3. Consider the complete graph K5 (a graph with 5 vertices where every pair of vertices is connected by an edge). Prove that K5 has a Hamiltonian circuit.

Eulerian Paths and Circuits

An Eulerian path is a path that traverses every edge exactly once. An Eulerian circuit is an Eulerian path that starts and ends at the same vertex.

4. Given the graph below, determine if an Eulerian path exists. If it does, find one.

Explore Hamiltonian and Eulerian paths and circuits with detailed problem-solving approaches. Learn how to determine if Hamiltonian paths, circuits, or Eulerian paths exist in graphs. Suitable for students in Grades 10-12, including proofs for K5 and step-by-step analysis for general graphs.

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