On almost hypohamiltonian graphs

TL;DR

This paper investigates almost hypohamiltonian graphs, providing existence results, bounds on their order, and classifications for planar and cubic cases, using specialized algorithms and extending previous theoretical results.

Contribution

It solves the open problem of the existence orders of a.h. graphs, introduces a specialized generation algorithm, and improves bounds on various graph properties.

Findings

Smallest cubic a.h. graphs have order 26.

Established bounds for planar a.h. graphs with specific girth.

Extended results on longest paths and cycles in polyhedral graphs.

Abstract

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.