Let be integers with , and set and . Because is quadratic in , there exists a such that A theorem by Erdős states that for , any -vertex nonhamiltonian graph with minimum degree has at most edges, and for the unique sharpness example is simply the graph . Erdős also presented a sharpness example for each .
We show that if and a -connected, nonhamiltonian -vertex graph with has more than edges, then is a subgraph of . Note that whenever .