Abstract: The varietal hypercube VQn, a variant of the hypercube Qn, was studied. It was proved that VQn contains a fault-free Hamilton cycle provided faulty edges do not exceed n－2, and that for two distinct vertices, x and y, there is a fault-free xy-Hamilton path in VQn provided faulty edges do not exceed n－3 for n≥3. The proof is based on an inductive construction.

Key words: graphs, Hamilton path, Hamilton cycle, varietal hypercube, fault-tolerant networks