{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,5]],"date-time":"2026-03-05T03:06:23Z","timestamp":1772679983873,"version":"3.50.1"},"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>For a graph $G = (V, E)$, a\u00a0fractional $[a, b]$-factor\u00a0is a real valued function $h:E(G)\\to [0,1]$ that satisfies\u00a0$a \\le ~ \\sum_{e\\in E_G(v)} h(e) ~ \\le b$ for all $ v\\in V(G)$, where $a$ and $b$ are real numbers and $E_G(v)$ denotes the set of edges incident with $v$.\u00a0In this paper, we prove that the condition $\\mathit{iso}(G-S) \\le (k+\\frac{1}{2})|S|$ is equivalent to the existence of fractional $[1,k+ \\frac{1}{2}]$-factors, where ${\\mathit{iso}}(G-S)$ denotes the number of isolated vertices in $G-S$. Using fractional factors as a tool, we construct component factors under the given isolated conditions. Namely, (i) a graph $G$ has a $\\{P_2,C_3,P_5, \\mathcal{T}(3)\\}$-factor if and only if $\\mathit{iso}(G-S) \\le \\frac{3}{2}|S|$ for all $S\\subset V(G)$;\u00a0(ii) a graph $G$ has a $\\{K_{1,1}, K_{1,2}, \\ldots,$ $K_{1,k}, \\mathcal{T}(2k+1)\\}$-factor ($k\\ge 2$)\u00a0if and only if $\\mathit{iso}(G-S) \\le (k+\\frac{1}{2})|S|$ for all $S\\subset V(G)$,\u00a0where $\\mathcal{T}(3)$ and $\\mathcal{T}(2k+1)$ are two special families of trees.<\/jats:p>","DOI":"10.37236\/8498","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T06:01:59Z","timestamp":1578636119000},"source":"Crossref","is-referenced-by-count":10,"title":["Fractional Factors, Component Factors and Isolated Vertex Conditions in Graphs"],"prefix":"10.37236","volume":"26","author":[{"given":"Roger","family":"Yu","sequence":"first","affiliation":[]},{"given":"Mikio","family":"Kano","sequence":"additional","affiliation":[]},{"given":"Hongliang","family":"Lu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2019,11,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i4p33\/7958","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i4p33\/7958","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:01:19Z","timestamp":1579233679000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i4p33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,11,22]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2019,10,11]]}},"URL":"https:\/\/doi.org\/10.37236\/8498","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,11,22]]},"article-number":"P4.33"}}