To solve this problem, let’s first define an Euler path. An Euler path is a path in a graph that visits every edge exactly once. Such a path exists if and only if the graph has exactly zero or two vertices with an odd degree.

Step 1: Determine the degree of each vertex.

From the graph:

• Vertex A is connected to B, C, and D (degree = 3, odd).
• Vertex B is connected to A and C (degree = 2, even).
• Vertex C is connected to A, B, and D (degree = 3, odd).
• Vertex D is connected to A and C (degree = 2, even).

There are two vertices with odd degrees: A and C. Thus, an Euler path exists, and it must start at one of these odd-degree vertices and end at the other.

Step 2: Find the Euler Path.

The Euler path will start at A (or C) and visit all edges exactly once, ending at C (or A). By following the edges systematically, one possible Euler path is:

A → B → C → D → A → C

Final Answer:

The sequence of vertices is: A, B, C, D, A, C

Let me know if you need more clarification or details on how I derived this! Here are some related questions for practice:

1. How do you determine whether a graph has an Euler circuit instead of an Euler path?
2. Can a graph have both an Euler path and an Euler circuit? Explain.
3. What happens if more than two vertices have an odd degree?
4. How does the concept of connected graphs affect Euler paths?
5. What is the difference between a Hamiltonian path and an Euler path?

Tip: Always check the degree of all vertices when solving for Euler paths—this is the key to knowing if a solution exists.