A graph is \emph{hamiltonian-connected} if every pair of vertices can be connected by a hamiltonian path, and it is \emph{hamiltonian} if it contains a hamiltonian cycle. We construct families of non-hamiltonian graphs for which the ratio of pairs of vertices connected by hamiltonian paths to all pairs of vertices approaches 1. We then consider minimal graphs that are hamiltonian-connected. It is known that any order-$n$ graph that is hamiltonian-connected must have $\geq 3n/2$ edges. We construct an infinite family of graphs realizing this minimum.

v3: substantial re-writing, including new author. To appear in Involve. v2: 12 pages, 6 figures. Substantial re-write including new results and removing results already proven by others. v1: 16 pages, 7 figures