{"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":1753893869923,"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>In this article we prove the following: Let $G$ be a $2$-connected graph with circumference $c(G)$. If\u00a0 $c(G)\\leq 5$, then $G$ has a spanning trail starting from any vertex, if\u00a0 $c(G)\\leq 7$, then $G$ has a spanning trail.\r\n\u00a0As applications of\u00a0 this result, we obtain the following.\r\n\r\n\r\nEvery $2$-edge-connected graph of order at most 8 has a spanning trail starting from any vertex\u00a0 with the exception of six graphs.\r\n\r\n\r\n\u00a0Let $G$ be a $2$-edge-connected graph and $S$ a subset of $V(G)$ such that $E(G-S)=\\emptyset$ and $|S|\\leq 6$. Then $G$ has a trail traversing all vertices of $S$ with the exception of two graphs, moreover, if $|S|\\leq 4$, then $G$ has a trail starting from any vertex of $S$ and containing $S$.\r\n\r\nEvery $2$-connected claw-free graph $G$ with order $n$ and minimum degree $\\delta(G)&gt; \\frac{n}{7}+4\\geq 23$ is traceable or belongs to two exceptional families of well-defined\u00a0 graphs, and moreover, if $\\delta(G)&gt; \\frac{n}{6}+4\\geq 13$, then $G$ is traceable.\r\n\r\nAll above results are sharp in a sense.<\/jats:p>","DOI":"10.37236\/8121","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T01:44:58Z","timestamp":1578620698000},"source":"Crossref","is-referenced-by-count":1,"title":["Spanning Trails in a 2-Connected Graph"],"prefix":"10.37236","volume":"26","author":[{"given":"Shipeng","family":"Wang","sequence":"first","affiliation":[]},{"given":"Liming","family":"Xiong","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,9,27]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p56\/7917","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i3p56\/7917","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:08:36Z","timestamp":1579216116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i3p56"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,9,27]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2019,7,4]]}},"URL":"https:\/\/doi.org\/10.37236\/8121","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,9,27]]},"article-number":"P3.56"}}