{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:00Z","timestamp":1753893840163,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $G$ be a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two and let $cc(G)$ be the length of its shortest cycle cover. It is known that if $cc(G) &lt; 1.4m$ in bridgeless graphs with $n_2 \\le m\/10$, then the Cycle Double Cover Conjecture holds. Fan (2017)\u00a0 proved that if $n_2 = 0$, then $cc(G) &lt; 1.6258m$ and $cc(G) &lt; 1.6148m$ provided that $G$ is loopless; morever, if $n_2 \\le m\/30$, then $cc(G) &lt; 1.6467m$. We show that for a bridgeless multigraph with $m$ edges and $n_2$ vertices of degree two, $cc(G) &lt; 1.6148m + 0.0741n_2$. Therefore, if $n_2=0$, then $cc(G) &lt; 1.6148m$ even if $G$ has loops; if $n_2 \\le m\/30$, then $cc(G) &lt; 1.6173m$; and if $n_2 \\le m\/10$, then $cc(G) &lt; 1.6223|E(G)|$. Our improvement is obtained by randomizing Fan's construction.<\/jats:p>","DOI":"10.37236\/9284","type":"journal-article","created":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T05:51:11Z","timestamp":1605851471000},"source":"Crossref","is-referenced-by-count":0,"title":["Short Cycle Covers of Graphs with at Most 77% Vertices of Degree Two"],"prefix":"10.37236","volume":"27","author":[{"given":"Anna","family":"Kompi\u0161ov\u00e1","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert","family":"Lukot'ka","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2020,11,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p31\/8212","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i4p31\/8212","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,11,20]],"date-time":"2020-11-20T05:51:11Z","timestamp":1605851471000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i4p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,11,13]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,10,2]]}},"URL":"https:\/\/doi.org\/10.37236\/9284","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2020,11,13]]},"article-number":"P4.31"}}