Problem 902

3. Here is an inductive "proof" that "every maximal planar graph is a plane triangulation with minimum degree 3": The induction starts with $K^{4}$. In the inductive step, consider any maximal planar graph $G$ of order $n$, and the maximal planar graph $G^{\prime}$ of order $n+1$ obtained by adding a new vertex $v$ to $G$ in all possible ways. No matter how $G^{\prime}$ is obtained, $v$ must lie in a face of $G$, and by the induction hypothesis, the boundary of this face is a triangle. Since $G^{\prime}$ is maximally planar, $v$ must be connected to the three vertices of this triangle. Clearly, $G^{\prime}$ is another triangulation, and $\delta\left(G^{\prime}\right)=d(v)=3$.
(i) - Identify the error in the "proof".
(ii) Find a counterexample and explain what the "proof" overlooks.