  We show that every $k$-tree of toughness greater than $\frac{k}{3}$ is
Hamilton-connected for $k \geq 3$. (In particular, chordal planar graphs of
toughness greater than $1$ are Hamilton-connected.) This improves the result of
Broersma et al. (2007) and generalizes the result of B\"ohme et al. (1999).
  On the other hand, we present graphs whose longest paths are short. Namely,
we construct $1$-tough chordal planar graphs and $1$-tough planar $3$-trees,
and we show that the shortness exponent of the class is $0$, at most
$\log_{30}{22}$, respectively. Both improve the bound of B\"ohme et al.
Furthermore, the construction provides $k$-trees (for $k \geq 4$) of toughness
greater than $1$.
