Inspired by the question, where the greatest number of edges in a hamiltonian graph was asked, I have a similar question, but for hamilton paths.
How many edges can a simple undirected graph contain, if it has no hamilton path ?
It is clear that a graph can have $\binom {n-1}{2}$ edges without having a hamilton path (just take the complete graph with $n-1$ vertices and an isolated vertex). But does a graph with $\binom {n-1}{2}+1$ edges necessarily have a hamilton path ?
What if the graph has to be connected ? What is the maximum number of edges in this case ?