Let G be a 9-regular connected graph that remains connected after deletingany 7 edges. Prove that G has a perfect matching

To prove that G has a perfect matching, we can use Tutte's theorem. Tutte's theorem states that a graph G has a perfect matching if and only if for every subset U of V(G), the number of odd components of G - U is less than or equal to |U|.

A 9-regular graph means that every vertex in the graph is connected to 9 other vertices.

Given that G remains connected after deleting any 7 edges, this means that for any subset U of V(G), the number of odd components of G - U is less than or equal to 7.

Since every vertex in G is connected to 9 other vertices, the maximum size of U that can create an odd component in G - U is 9.

Therefore, for every subset U of V(G), the number of odd components of G - U is less than or equal to |U|.

By Tutte's theorem, G has a perfect matching.