  In an orientation $O$ of the graph $G$, the edge $e$ is deletable if and only
if $O-e$ is strongly connected. For a $3$-edge-connected graph $G$, H\"orsch
and Szigeti defined the Frank number as the minimum $k$ for which $G$ admits
$k$ orientations such that every edge $e$ of $G$ is deletable in at least one
of the $k$ orientations. They conjectured the Frank number is at most $3$ for
every $3$-edge-connected graph $G$. They proved the Petersen graph has Frank
number $3$, but this was the only example with this property. We show an
infinite class of graphs having Frank number $3$. H\"orsch and Szigeti showed
every $3$-edge-colorable $3$-edge-connected graph has Frank number at most $3$.
It is tempting to consider non-$3$-edge-colorable graphs as candidates for
having Frank number greater than $2$. Snarks are sometimes a good source of
finding critical examples or counterexamples. One might suspect various snarks
should have Frank number $3$. However, we prove several candidate infinite
classes of snarks have Frank number $2$. As well as the generalized Petersen
Graphs $GP(2s+1,s)$. We formulate numerous conjectures inspired by our
experience.
