Non-Hamiltonian 3-Regular Graphs with Arbitrary Girth

Abstract

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.