{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:16:28Z","timestamp":1759335388053,"version":"3.41.2"},"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>Let $G$ be a simple graph, and let $\\Delta(G)$ and $\\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\\chi'(G)=\\Delta(G)$ or $\\chi'(G)=\\Delta(G)+1$. We say $G$ is $\\Delta$-critical if $\\chi'(G)=\\Delta(G)+1$ and $\\chi'(H)&lt;\\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\\Delta$-critical graph, then\u00a0 $G$ has a 2-factor. Let $G$ be an $n$-vertex $\\Delta$-critical graph. It was proved that if $\\Delta(G)\\ge n\/2$, then $G$ has a 2-factor; and that if $\\Delta(G)\\ge 2n\/3+13$, then $G$\u00a0 has a hamiltonian cycle, and thus a 2-factor.\u00a0It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $\\Delta$-critical graph under \"moderate\" given toughness and\u00a0 maximum degree conditions. In particular, we show that\u00a0 if $G$ is an\u00a0 $n$-vertex $\\Delta$-critical graph with toughness at least 3\/2 and with maximum degree at least $n\/3$, then $G$ has a 2-factor. We also construct a family of graphs that have order $n$, maximum degree $n-1$, toughness at least $3\/2$, but have no 2-factor. This implies that the $\\Delta$-criticality in the result is needed.\u00a0In addition, we develop new techniques in proving the existence of 2-factors in graphs.<\/jats:p>","DOI":"10.37236\/7353","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T02:08:09Z","timestamp":1578622089000},"source":"Crossref","is-referenced-by-count":2,"title":["Vizing's 2-Factor Conjecture Involving Toughness and Maximum Degree  Conditions"],"prefix":"10.37236","volume":"26","author":[{"given":"Jinko","family":"Kanno","sequence":"first","affiliation":[]},{"given":"Songling","family":"Shan","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,5,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p17\/7831","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p17\/7831","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,16]],"date-time":"2020-01-16T23:14:48Z","timestamp":1579216488000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7353","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2019,5,3]]},"article-number":"P2.17"}}