Sheehan’s conjecture on multiple Hamiltonian cycles in 4-regular graphs

Prove that every 4-regular graph that contains a Hamiltonian cycle necessarily contains a second Hamiltonian cycle distinct from the first, or otherwise construct a counterexample.

Background

Regular graph generation is motivated in part by long-standing Hamiltonicity questions. Sheehan’s conjecture asserts non-uniqueness of Hamiltonian cycles in 4-regular Hamiltonian graphs, a problem connected to cycle structure and robust Hamiltonicity in regular graphs. Despite computational and theoretical advances, no general resolution is known.