  It is well known that 3--regular graphs with arbitrarily large girth exist.
Three constructions are given that use the former to produce non-Hamiltonian
3--regular graphs without reducing the girth, thereby proving that such graphs
with arbitrarily large girth also exist. The resulting graphs can be 1--, 2--
or 3--edge-connected depending on the construction chosen. From the
constructions arise (naive) upper bounds on the size of the smallest
non-Hamiltonian 3--regular graphs with particular girth. Several examples are
given of the smallest such graphs for various choices of girth and
connectedness.
