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A (k, g) graph is a graph with regular degree k and girth g. The cage problemrefers to finding the smallest (k, g) graph. The (3, 14) cage problem is knownto 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 Hamiltonianbipartite class of trivalent graphs. A parameter called symmetry factor forrepresenting rotational symmetry is introduced in this paper. The generalproblem of finding a (3, g) Hamiltonian bipartite graph of minimum order isfurther decomposed into a set of sub-problems for finding (3, g) Hamiltonianbipartite graphs of minimum order for various symmetry factors. The minimumorder for (3, g) Hamiltonian bipartite graphs for various symmetry factors hasbeen found using computer search. This information about sub-problems alsoyields useful information about non-existence of (3, 14) Hamiltonian bipartitegraphs 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.

Sub-problems of the (3, 14) cage problem and their computer analysis