{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:29Z","timestamp":1753893869500,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a graph whose vertices are labeled $1,\\ldots,n$, and $\\pi$ be a permutation on $[n]:=\\{1,2,\\ldots, n\\}$. A pebble $p_i$ that is initially placed at the vertex $i$ has destination $\\pi(i)$ for each $i\\in [n]$. At each step, we choose a matching and swap the two pebbles on each of the edges. Let $rt(G, \\pi)$, the routing number for $\\pi$, be the minimum number of steps necessary for the pebbles to reach their destinations.Li, Lu and Yang proved that $rt(C_n, \\pi)\\le n-1$ for every permutation $\\pi$ on the $n$-cycle $C_n$ and conjectured that for $n\\geq 5$, if $rt(C_n, \\pi) = n-1$, then $\\pi = 23\\cdots n1$ or its inverse. By a computer search, they showed that the conjecture holds for $n&lt;8$. We prove in this paper that the conjecture holds for all even $n\\ge 6$.<\/jats:p>","DOI":"10.37236\/5422","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:20:52Z","timestamp":1578669652000},"source":"Crossref","is-referenced-by-count":0,"title":["Extremal Permutations in Routing Cycles"],"prefix":"10.37236","volume":"23","author":[{"given":"Jinhua","family":"He","sequence":"first","affiliation":[]},{"given":"Louis A.","family":"Valentin","sequence":"additional","affiliation":[]},{"given":"Xiaoyan","family":"Yin","sequence":"additional","affiliation":[]},{"given":"Gexin","family":"Yu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,9,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p47\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i3p47\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T00:16:38Z","timestamp":1579220198000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i3p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,16]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2016,7,8]]}},"URL":"https:\/\/doi.org\/10.37236\/5422","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,9,16]]},"article-number":"P3.47"}}