  A (k, g) graph is a graph with regular degree k and girth g. The cage problem
refers to finding the smallest (k, g) graph. The (3, 14) cage problem is known
to be unresolved. In 2002, Exoo found a (3, 14) record graph with order 384.
The trivalent cage problem is restricted in this paper to the Hamiltonian
bipartite class of trivalent graphs. A parameter called symmetry factor for
representing rotational symmetry is introduced in this paper. The general
problem of finding a (3, g) Hamiltonian bipartite graph of minimum order is
further decomposed into a set of sub-problems for finding (3, g) Hamiltonian
bipartite graphs of minimum order for various symmetry factors. The minimum
order for (3, g) Hamiltonian bipartite graphs for various symmetry factors has
been found using computer search. This information about sub-problems also
yields useful information about non-existence of (3, 14) Hamiltonian bipartite
graphs between the (3, 14) lower bound, 258 and the (3, 14) upper bound, 384.
This non-existence information partially supports the likelihood of the current
(3, 14) record graph indeed being the (3, 14) cage.
