The question asks to determine the Euler's path and Euler's circuit for the given graph, and provides multiple-choice answers. Let us analyze the graph to solve the problem.

Key Definitions:

1. Euler's Path: A path that visits every edge of a graph exactly once. It exists if the graph has exactly 0 or 2 vertices of odd degree.
2. Euler's Circuit: A circuit that visits every edge of a graph exactly once and starts and ends at the same vertex. It exists if all vertices have even degrees.

Step 1: Degree of Each Vertex

From the graph:

• Vertex A has degree 2 (connected to B and D).
• Vertex B has degree 3 (connected to A, C, and itself in the loop).
• Vertex C has degree 2 (connected to B and D).
• Vertex D has degree 3 (connected to A, B, and C).

Step 2: Check for Euler Path or Circuit

• Since B and D have odd degrees, there are exactly 2 vertices of odd degree. Thus, an Euler's Path exists, starting at one odd-degree vertex and ending at the other.
• However, since not all vertices have even degrees, no Euler's Circuit exists.

Step 3: Answer Choices

Given the graph's structure:

• Euler's Path = CABBCDA is correct.
• Euler's Circuit = None is correct (as explained above).

The correct answer to the multiple-choice question is:

• Euler's Path = CABBCDA and Euler's Circuit = None.

Would you like a detailed walkthrough or verification of another part? Let me know!

Related Questions:

1. What is the difference between a Hamiltonian path and an Eulerian path?
2. How do you find the degree of vertices in a multigraph?
3. Why is it important to check for odd-degree vertices in Eulerian graphs?
4. Can a graph with 4 odd-degree vertices have an Eulerian path or circuit?
5. What modifications can be made to convert the given graph into one with an Eulerian circuit?

Tip:

Always count the degree of each vertex carefully to determine the existence of Euler's paths or circuits.