{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:49Z","timestamp":1753893829001,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>We show that, in a $k$-connected graph $G$ of order $n$ with $\\alpha(G) =  \\alpha$, between any pair of vertices, there exists a path $P$ joining them  with $$|P| \\geq \\min \\left\\{ n, \\frac{(k - 1)(n - k)}{\\alpha} + k \\right\\}.$$   This implies that, for any edge $e \\in E(G)$, there is a cycle containing $e$  of length at least $$\\min \\left\\{ n, \\frac{(k - 1)(n - k)}{\\alpha} + k  \\right\\}.$$  Moreover, we generalize our result as follows: for any choice  $S$ of $s \\leq k$ vertices in $G$, there exists a tree $T$ whose set of leaves is  $S$ with $$|T| \\geq \\min \\left\\{ n, \\frac{(k - s + 1)(n - k)}{\\alpha} + k  \\right\\}.$$<\/jats:p>","DOI":"10.37236\/636","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:43:51Z","timestamp":1578696231000},"source":"Crossref","is-referenced-by-count":1,"title":["Long Path Lemma concerning Connectivity and Independence Number"],"prefix":"10.37236","volume":"18","author":[{"given":"Shinya","family":"Fujita","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Halperin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Colton","family":"Magnant","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,7,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p149\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p149\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:07:54Z","timestamp":1579284474000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p149"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,7,22]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/636","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2011,7,22]]},"article-number":"P149"}}