I have a question regarding the concept of Eulerian and Hamiltonian graphs, and I need assistance in comprehending the proof provided for a specific problem.
The problem: We are asked to justify, using theorems, whether or not the given graph is an Eulerian graph and/or a Hamiltonian graph.
Solution: The solution states that the graph is not an Eulerian graph due to having vertices with odd degrees. Then, it references Ores's theorem (?) and asserts that if all vertices have degrees greater than or equal to the number of vertices divided by 2, the graph is a Hamiltonian graph.
However, I'm having difficulty understanding the proof and why it concludes that the graph is a Hamiltonian graph but not an Eulerian graph. I need help in breaking down the logic behind this proof and understanding the application of the theorems involved.