{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T17:01:17Z","timestamp":1776963677443,"version":"3.51.4"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>There is a rich history of studying the existence of cycles in planar graphs. The famous Tutte theorem on the Hamilton cycle states that every 4-connected planar graph contains a Hamilton cycle. Later on, Thomassen (1983), Thomas and Yu (1994) and Sanders (1996) respectively proved that every 4-connected planar graph contains a cycle of length $n-1, n-2$ and $n-3$. Chen, Fan and Yu (2004) further conjectured that every 4-connected planar graph contains a cycle of length $\\ell$ for $\\ell\\in\\{n,n-1,\\ldots,n-25\\}$ and they verified that for $\\ell\\in \\{n-4, n-5, n-6\\}$. When we remove the ``4-connected\" condition, how do we guarantee the existence of a long cycle in a planar graph? A natural question asks by adding a spectral radius condition: What is the smallest real number $C(n)$ such that for sufficiently large $n$, every planar graph $G$ of order $n$ with spectral radius $\\rho(G)$ greater than $C(n)$ contains a long cycle? In this paper, we give a stronger answer to the above question. Let $G$ be a planar graph with order $n\\geq 1.8\\times 10^{17}$ and $k\\leq \\lfloor\\log_2(n-3)\\rfloor-8$ be a non-negative integer, we show that if $\\rho(G)\\geq \\rho(K_2\\vee(P_{n-2k-4}\\cup 2P_{k+1}))$ then $G$ contains a cycle of length $\\ell$ for every $\\ell\\in \\{n-k, n-k-1, \\ldots, 3\\}$ unless $G\\cong K_2\\vee(P_{n-2k-4}\\cup 2P_{k+1})$.<\/jats:p>","DOI":"10.37236\/13075","type":"journal-article","created":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T19:56:26Z","timestamp":1747166186000},"source":"Crossref","is-referenced-by-count":0,"title":["Long Cycles and Spectral Radii in Planar Graphs"],"prefix":"10.37236","volume":"32","author":[{"given":"Ping","family":"Xu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lin","family":"Huiqiu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Longfei","family":"Fang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2025,5,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p26\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v32i2p26\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T19:56:26Z","timestamp":1747166186000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v32i2p26"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,13]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,4,11]]}},"URL":"https:\/\/doi.org\/10.37236\/13075","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,13]]},"article-number":"P2.26"}}