Constructing hypohamiltonian snarks with cyclic connectivity 5 and 6 (Q870078)

Summary: A graph is hypohamiltonian if it is not hamiltonian but every vertex-deleted subgraph is. In this paper we study hypohamiltonian snarks -- cubic hypohamiltonian graphs with chromatic index 4. We describe a method, based on superposition of snarks, which produces new hypohamiltonian snarks from old ones. By choosing suitable ingredients we can achieve that the resulting graphs are cyclically 5-connected or 6-connected. So far, only three sporadic hypohamiltonian snarks with cyclic connectivity 5 have been found, while the flower snarks of Isaacs constitute the only known family of cyclically 6-connected hypohamiltonian snarks. Our method produces hypohamiltonian snarks with cyclic connectivity 5 and 6 for all but finitely many even orders.