  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.
