1. **Problem Statement:** Determine if the given graphs have an Euler circuit based on their vertex degrees. 2. **Key Concept:** A graph has an Euler circuit if and only if it is connected and every vertex has an even degree. 3. **Part (a):** Degrees are 2, 2, 3, 3, and 4. - The graph is connected. - Vertices with degrees 3 are odd. - Since not all vertices have even degree, the graph does **not** have an Euler circuit. 4. **Part (b):** Degrees are 2, 2, 4, 4, and 6. - The graph is connected. - All vertices have even degrees. - Therefore, the graph **does** have an Euler circuit. 5. **Part (c):** Degrees are 2, 2, 4, 4, and 6. - The graph is not necessarily connected. - Even though all degrees are even, without connectivity, an Euler circuit is **not necessarily** present. **Final answers:** - (a) No - (b) Yes - (c) Not necessarily