1. **Problem Statement:** Determine if the given graph has (i) an Euler's path and (ii) an Euler's circuit. 2. **Recall Definitions:** - An **Euler's path** is a trail in a graph that visits every edge exactly once. - An **Euler's circuit** is an Euler's path that starts and ends on the same vertex. 3. **Key Theorems:** - A connected graph has an Euler's circuit if and only if every vertex has an even degree. - A connected graph has an Euler's path (but not an Euler's circuit) if and only if exactly two vertices have odd degree. 4. **Analyze Vertex Degrees:** - Vertex 1 connects to 2 and 3 → degree 2 (even) - Vertex 2 connects to 1 and 3 → degree 2 (even) - Vertex 3 connects to 1, 2, 4, 6 → degree 4 (even) - Vertex 4 connects to 3 and 5 → degree 2 (even) - Vertex 5 connects to 4 and 6 → degree 2 (even) - Vertex 6 connects to 3, 5, 7 → degree 3 (odd) - Vertex 7 connects to 6 → degree 1 (odd) 5. **Count Odd Degree Vertices:** - Odd degree vertices are 6 and 7 (2 vertices). 6. **Conclusion:** - Since exactly two vertices have odd degree, the graph has an Euler's path but does not have an Euler's circuit. **Final answer:** - (i) Euler's path: Yes, because exactly two vertices have odd degree. - (ii) Euler's circuit: No, because not all vertices have even degree.