New results on hypohamiltonian and almost hypohamiltonian graphs.
Jan Goedgebeur, Carol T. Zamfirescu · arXiv (Cornell University) · 2016
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 $v e w$. 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.