7,165 research outputs found
On almost hypohamiltonian graphs
A graph is almost hypohamiltonian (a.h.) if is non-hamiltonian, there
exists a vertex in such that is non-hamiltonian, and is
hamiltonian for every vertex in . 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.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717
- …