New results on hypohamiltonian and almost hypohamiltonian graphs

Consider a non-hamiltonian graph G. G is hypohamiltonian if for every vertex v in G, the graph G − v is hamiltonian. G is almost hypohamiltonian if there exists a vertex w in G such that G−w is non-hamiltonian, and G− v is hamiltonian for every vertex v 6= w. McKay asked in [J. Graph Theory, doi: 10.1002/jgt.22043] whether infinitely many planar cubic hypohamiltonian graphs of girth 5 exist. We settle this question affirmatively. The second author asked in [J. Graph Theory 79 (2015) 63–81] to find all orders for which almost hypohamiltonian graphs exist—we solve this problem as well. To this end, we present a specialised algorithm which generates complete sets of almost hypohamiltonian graphs for various orders. Furthermore, we show that the smallest cubic almost hypohamiltonian graphs have order 26. We also provide a lower bound for the order of the smallest planar almost hypohamiltonian graph and improve the upper bound for the order of the smallest planar almost hypohamiltonian graph containing a cubic vertex. Finally, we determine the smallest planar almost hypohamiltonian graphs of girth 5, both in the general and cubic case.