## Jan Goedgebeur ; Carol T. Zamfirescu - On almost hypohamiltonian graphs

dmtcs:5300 - Discrete Mathematics & Theoretical Computer Science, July 30, 2019, vol. 21 no. 4 - https://doi.org/10.23638/DMTCS-21-4-5
On almost hypohamiltonian graphs

Authors: Jan Goedgebeur ; Carol T. Zamfirescu

A graph $G$ is almost hypohamiltonian (a.h.) if $G$ is non-hamiltonian, there exists a vertex $w$ in $G$ such that $G - w$ is non-hamiltonian, and $G - v$ is hamiltonian for every vertex $v \ne w$ in $G$. The second author asked in [J. Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.

Volume: vol. 21 no. 4
Section: Graph Theory
Published on: July 30, 2019
Accepted on: July 30, 2019
Submitted on: March 21, 2019
Keywords: Mathematics - Combinatorics,Computer Science - Discrete Mathematics