Think through the following proof: We will show that any simple graph where every vertex has degree at least 1 is connected. As a base case, we have two vertices connected by a single edge. Now, suppose that for n ≤ k, a simple graph with n vertices, each of which has degree at least 1, is connected. Consider a simple graph G with k vertices, each of which has degree at least 1. By the inductive hypothesis, it is connected. Add a vertex v to G so that we have G-with-v, which has k + 1 vertices; in order that every vertex has degree at least 1, we also have to add an edge to v. But an edge in a simple graph must connect two vertices, so the other end of the edge must be incident to a vertex of G. Thus, G-with-v is connected. What’s wrong with this proof? It can’t be right—consider the graph in Figure 4.11.