Systematic searches for hypohamiltonian graphs

A graph G is called hypohamiltonian if G is not hamiltonian but every vertex-deleted subgraph G-v is hamiltonian. The existence of a p-vertex hypohamiltonian graph is open only for p = 14,17, and the existence of a p-vertex, cubic hypohamiltonian graph is open only for p = 14,16,24,32. With the aid of a computer we have established that there is no hypohamiltonian graph of order 14, and no cubic hypohamiltonian graph of order 14 or 16. In addition, there is no hypohamiltonian graph of order 17 with girth ≥ 5. On the other hand, new p-vertex, cubic hypohamiltonian graphs have been found for p = 18,22.

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