Hey guy i am soooo stuck on this question any help would be greatly appreciated!!

Let G be a 9-regular graph with $\displaystyle p$ vertices and suppose $\displaystyle G$ has the property that any subgraph with more that $\displaystyle \frac{p}{2} - 1$ edges has a vertex of degree at least 2. Prove that there is a subgraph $\displaystyle H$ of $\displaystyle G$ with the property that if the vertices of $\displaystyle H$ and all the edges incident with the vertices of $\displaystyle H$ are removed from $\displaystyle G$, then what remains is a graph with more than $\displaystyle |V(H)| + 1$components of odd order.