Constructing Hypohamiltonian Snarks with Cyclic Connectivity $5$ and $6$

Abstract

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. Previously, only three sporadic hypohamiltonian snarks with cyclic connectivity 5 had been found, while the flower snarks of Isaacs were 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.