We must be in the same class. Have you worked out the answer yet? I've been working on it all day without any luck.
Let G be a 9-regular graph with p vertices and suppose G has the property that any subgraph with more that p/2 − 1 edges has a vertex of degree at least 2. Prove that there is a subgraph H of G with the property that if the vertices of H and all the edges incident with the vertices of H are removed from G, then what remains is a graph with more than |V (H)| + 1
components of odd order.