{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:39:04Z","timestamp":1787323144658,"version":"build-2736575974"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 1954134"],"award-info":[{"award-number":["DMS 1954134"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Discrete Math."],"published-print":{"date-parts":[[2021,1]]},"abstract":"<jats:p>Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices, then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n\/{\\log}_2 n)$ separating 4-cycles, then $G$ has $\\Omega(n^2)$ Hamiltonian cycles, and if $\\delta(G)\\ge 5$, then $G$ has $2^{\\Omega(n^{1\/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a \u201cdouble wheel\u201d structure, providing further evidence to the above conjecture.<\/jats:p>","DOI":"10.1137\/20m1366551","type":"journal-article","created":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T11:57:11Z","timestamp":1621252631000},"page":"1005-1021","source":"Crossref","is-referenced-by-count":3,"title":["Number of Hamiltonian Cycles in Planar Triangulations"],"prefix":"10.1137","volume":"35","author":[{"given":"Xiaonan","family":"Liu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xingxing","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,5,17]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2019.04.005"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0118(199909)32:1<81::AID-JGT8>3.0.CO;2-9"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.22149"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190030407"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.10057"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.disc.2020.112126"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1006\/jctb.1994.1058"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190070205"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1956-0081471-8"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.2307\/1968197"}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1366551","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:44:16Z","timestamp":1787319856000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1366551"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":10,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1366551"],"URL":"https:\/\/doi.org\/10.1137\/20m1366551","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}