A Hamiltonian graph is a graph which has a Hamiltonian cycle.
A Hamiltonian cycle is a cycle which crosses all of the vertices of a graph. According to Ore's theorem , if $p \ge 3$ we have this :
For each two non-adjacent vertices $u,v$ , if $\deg(u)+\deg(v) \ge p$, then the graph is Hamiltonian.
Now suppose that we have a graph with $p$ vertices and $2+(p-1)(p-2)/2$ edges. How can we prove that this graph is Hamiltonian ?