On a conjecture of Faudree and Schelp

In 1976 Faudree and Schelp conjectured that in a hamiltonian-connected graph on $n$ vertices, any two distinct vertices are connected by a path of length $k$ for every $k \ge n/2$. In 1978 Thomassen constructed a (non-cubic and non-planar) family of counterexamples, showing that there exist hamiltonian-connected $n$-vertex graphs containing two vertices with no path of length $n-2$ between them. We complement this result by describing cubic planar counterexamples on $6p+16$ vertices, each containing vertices between which there is no path of any odd length greater than $1$ and at most $4p+9$. Motivated by a remark of Thomassen about a gap in the cycle spectrum of hamiltonian-connected graphs, we also describe an infinite family of hamiltonian-connected graphs with many gaps in the first half of their cycle spectra.