1. The problem statement, all variables and given/known data I am trying to verify this theorem for myself. Theorem: Every graph G containing a cycle satisfies g(G)≤2diam(G)+1 2. Relevant equations N/A 3. The attempt at a solution i have drawn graphs and i failed to verify the theorem. is it even possible, since increasing the girth directly increases the diameter of the graph...? However the theorem has a proof. Proof: Let C be a shortest cycle in G. If g(G)>= 2diam(G) + 2, then C has two vertices whose distance in C is at least diam(G)+1. In G, these vertices have a lesser distance; any shortest path P between them is therefore not a subgraph of C. Thus, P contains a C-path xPy. Together with the shorter of the two x-y paths in C, this path xPy forms a shorter cycle than C, a contradiction.