consider the following graph.

(a) Without adding new vertices, add a single edge to the graph so that the new graph will have an Euler path. Which vertices does your new edge connect?
a and h

To ensure the graph has an Euler path, we need to remember the condition for such a path: **A graph will have an Euler path if it has exactly two vertices of odd degree**.

### Step-by-step approach:

1. **Check the degree of each vertex**:  
   Start by identifying which vertices have an odd number of edges (odd degree) and which have an even number of edges (even degree).
   
2. **Condition for an Euler path**:  
   An Euler path exists if exactly two vertices have an odd degree.

3. **Connecting two odd-degree vertices**:  
   If there are more than two vertices with an odd degree, we need to add an edge between two of them to reduce the number of odd-degree vertices to exactly two.

4. **Why connect vertices a and h?**:  
   Adding an edge between two odd-degree vertices (like \(a\) and \(h\)) will reduce their degrees by 1, possibly making them even. This adjustment would then satisfy the condition for an Euler path.

Thus, the edge you should add connects vertices **a and h**. After adding this edge, you should verify that exactly two vertices still have odd degrees.

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Do you want to review the graph's vertex degrees, or have any questions on this process?

### 5 Related Questions:
1. What is the difference between an Euler path and an Euler circuit?
2. How do we determine if a graph has an Euler circuit?
3. Can a graph have more than one Euler path?
4. What is the importance of odd and even degree vertices in graph theory?
5. What other methods can be used to make a graph Eulerian?

**Tip:** When analyzing graphs, always first check the degrees of the vertices. It's a quick way to identify if a path or circuit is possible.

Without adding new vertices, add a single edge to the graph so that the new graph will have an Euler path. Which vertices does your new edge connect? a and h

Learn how to add a single edge between vertices a and h in a graph to create an Euler path. This problem explores graph theory concepts, such as Euler paths, vertex degrees, and Euler's theorem. Suitable for students in Grades 10-12.

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