{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:40:56Z","timestamp":1753890056466,"version":"3.41.2"},"reference-count":0,"publisher":"Centre pour la Communication Scientifique Directe (CCSD)","issue":"Graph Theory","license":[{"start":{"date-parts":[[2019,7,30]],"date-time":"2019-07-30T00:00:00Z","timestamp":1564444800000},"content-version":"am","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,7,30]],"date-time":"2019-07-30T00:00:00Z","timestamp":1564444800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"},{"start":{"date-parts":[[2019,7,30]],"date-time":"2019-07-30T00:00:00Z","timestamp":1564444800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/arxiv.org\/licenses\/nonexclusive-distrib\/1.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"accepted":{"date-parts":[[2025,3,31]]},"abstract":"<jats:p>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, {\\&amp;quot;O}sterg{\\aa}rd, Pettersson, and the second author.<\/jats:p><jats:p>Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.07171<\/jats:p>","DOI":"10.23638\/dmtcs-21-4-5","type":"journal-article","created":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:56:12Z","timestamp":1743699372000},"source":"Crossref","is-referenced-by-count":0,"title":["On almost hypohamiltonian graphs"],"prefix":"10.23638","volume":"vol. 21 no. 4","author":[{"given":"Jan","family":"Goedgebeur","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carol T.","family":"Zamfirescu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"25203","published-online":{"date-parts":[[2019,7,30]]},"container-title":["Discrete Mathematics &amp; Theoretical Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/arxiv.org\/pdf\/1606.06577v4","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/arxiv.org\/pdf\/1606.06577v4","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,3]],"date-time":"2025-04-03T16:56:12Z","timestamp":1743699372000},"score":1,"resource":{"primary":{"URL":"http:\/\/dmtcs.episciences.org\/5300"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,30]]},"references-count":0,"journal-issue":{"issue":"Graph Theory","published-online":{"date-parts":[[2019,7,30]]}},"URL":"https:\/\/doi.org\/10.23638\/dmtcs-21-4-5","relation":{"has-preprint":[{"id-type":"arxiv","id":"1606.06577v3","asserted-by":"subject"}],"is-same-as":[{"id-type":"arxiv","id":"1606.06577","asserted-by":"subject"},{"id-type":"doi","id":"10.48550\/arXiv.1606.06577","asserted-by":"subject"}]},"ISSN":["1365-8050"],"issn-type":[{"type":"electronic","value":"1365-8050"}],"subject":[],"published":{"date-parts":[[2019,7,30]]},"article-number":"5300"}}