{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T01:34:53Z","timestamp":1649036093837},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"Given two $k$-graphs $F$ and $H$, a perfect $F$-tiling (also called an $F$-factor) in $H$ is a set of vertex-disjoint copies of $F$ that together cover the vertex set of $H$. Let $t_{k-1}(n, F)$ be the smallest integer $t$ such that every\u00a0 $k$-graph $H$ on $n$ vertices with minimum codegree at least $t$ contains a perfect $F$-tiling.\u00a0 Mycroft (JCTA, 2016) determined\u00a0 the asymptotic values of $t_{k-1}(n, F)$ for $k$-partite $k$-graphs $F$ and conjectured that the error terms $o(n)$ in $t_{k-1}(n, F)$ can be replaced by a constant that depends only on $F$. In this paper, we determine the exact value of $t_2(n, K_{m,m}^{3})$, where $K_{m,m}^{3}$ (defined by Mubayi and Verstra\u00ebte, JCTA, 2004) is the 3-graph obtained from the complete bipartite graph $K_{m,m}$ by replacing each vertex in one part by a 2-elements set. Note that $K_{2,2}^{3}$ is\u00a0 the well known\u00a0 generalized 4-cycle $C_4^3$ (the 3-graph on six vertices and four distinct edges $A, B, C, D$ with $A\\cup B= C\\cup D$ and $A\\cap B=C\\cap D=\\emptyset$). The result confirms Mycroft's conjecture for $K_{m,m}^{3}$. Moreover, we improve the error term $o(n)$ to a sub-linear term when $F=K^3(m)$ and show that the sub-linear term is tight for $K^3(2)$, where $K^3(m)$ is the complete $3$-partite $3$-graph with each part of size $m$.<\/jats:p>","DOI":"10.37236\/9061","type":"journal-article","created":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:10Z","timestamp":1599187630000},"source":"Crossref","is-referenced-by-count":0,"title":["Codegree Threshold for Tiling Balanced Complete $3$-Partite $3$-Graphs and Generalized $4$-Cycles"],"prefix":"10.37236","volume":"27","author":[{"given":"Xinmin","family":"Hou","sequence":"first","affiliation":[]},{"given":"Boyuan","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Yue","family":"Ma","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2020,9,4]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p47\/8165","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v27i3p47\/8165","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T02:47:10Z","timestamp":1599187630000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v27i3p47"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,4]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2020,7,9]]}},"URL":"http:\/\/dx.doi.org\/10.37236\/9061","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,4]]}}}
