Hypohamiltonian Snarks with Cyclic Connectivity 5 and 6

A non-hamiltonian graph is called hypohamiltonian if it becomes hamiltonian by deleting an arbitrary vertex. In this paper we describe a construction which produces cyclically 5-connected and cyclically 6-connected hypohamiltonian snarks (cubic graphs with chromatic index 4) of all but finitely many orders. Our construction considerably extends the collection of known hypohamilonian cubic graphs.