{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:45:21Z","timestamp":1787323521497,"version":"build-2736575974"},"reference-count":2,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"publisher","award":["NS2020055"],"award-info":[{"award-number":["NS2020055"]}],"id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971406"],"award-info":[{"award-number":["11971406"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11622110"],"award-info":[{"award-number":["11622110"]}],"id":[{"id":"10.13039\/501100001809","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>For any positive integer $k$, define $f(k)$ (respectively, $f_3(k)$) to be the minimal integer $\\ge k$ such that every 3-connected planar graph $G$ (respectively, 3-connected cubic planar graph $G$) of circumference $\\ge k$ has a cycle whose length is in the interval $[k, f(k)]$ (respectively, $[k, f_3(k)]$). Merker showed that $f_3(k) \\le 2k + 9$ for any $k \\ge 2$, and $f_3(k) \\ge 2k + 2$ for any even $k \\ge 4$. He conjectured that $f_3(k) \\le 2k + 2$ for any $k \\ge 2$. This conjecture was disproved by Zamfirescu, who gave an infinite family of counterexamples for every even $k \\ge 6$ whose graphs have no cycle length in $[k, 2k + 2]$, i.e., $f_3(k) \\ge 2k + 3$ for any even $k \\ge 6$. However, the exact value of $f_3(k)$ was only known for $k \\le 4$, and it is a natural problem to determine $f_3(k)$ for $k \\ge 5$. In this paper, we improve Merker's upper bound, and give the exact value of $f_3(k)$ for every $k \\ge 5$. We show that $f_3(5) = 10$, $f_3(7) = 15$, $f_3(9) = 20$, and $f_3(k) = 2k + 3$ for any $k = 6, 8$ or $\\ge 10$. For general 3-connected planar graphs, Merker conjectured that there exists some positive integer $c$ such that $f(k) \\le 2k + c$ for any positive integer $k$. We give a complete positive answer to this conjecture. We prove that $f(k) = 5$ for any $k \\le 3$, $f(4) = 10$, and $f(k) = 2k + 3$ for any $k \\ge 5$.<\/jats:p>","DOI":"10.1137\/20m1366770","type":"journal-article","created":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T12:47:21Z","timestamp":1630586841000},"page":"2039-2048","source":"Crossref","is-referenced-by-count":4,"title":["Tight Gaps in the Cycle Spectrum of 3-Connected Planar Graphs"],"prefix":"10.1137","volume":"35","author":[{"given":"Qing","family":"Cui","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8691-7749","authenticated-orcid":true,"given":"On-Hei","family":"Solomon Lo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2021,9,2]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jctb.2020.08.002"},{"key":"atypb2","unstructured":"C. T. Zamfirescu,\n                      Counterexamples to a Conjecture of Merker on 3-Connected Cubic Planar Graphs with a Large Cycle Spectrum Gap\n                      , preprint,https:\/\/arxiv.org\/abs\/2009.00423(2020)."}],"container-title":["SIAM Journal on Discrete Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1366770","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:07:13Z","timestamp":1787321233000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1366770"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1]]},"references-count":2,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,1]]}},"alternative-id":["10.1137\/20M1366770"],"URL":"https:\/\/doi.org\/10.1137\/20m1366770","relation":{},"ISSN":["0895-4801","1095-7146"],"issn-type":[{"value":"0895-4801","type":"print"},{"value":"1095-7146","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1]]}}}