{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T16:25:08Z","timestamp":1759335908327,"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>For $1\\leqslant \\ell&lt; k$,\u00a0 an $\\ell$-overlapping $k$-cycle\u00a0is a $k$-uniform hypergraph in which,\u00a0for some cyclic vertex ordering, every edge consists of $k$ consecutive vertices and every two\u00a0consecutive edges share exactly $\\ell$ vertices.A $k$-uniform hypergraph $H$ is $\\ell$-Hamiltonian saturated\u00a0if $H$ does not contain an\u00a0$\\ell$-overlapping Hamiltonian $k$-cycle but every hypergraph obtained from $H$ by adding one edge\u00a0does contain such a cycle. Let $\\mathrm{sat}(n,k,\\ell)$ be the smallest number of edges in an\u00a0$\\ell$-Hamiltonian saturated $k$-uniform hypergraph on $n$ vertices. In the case of graphs Clark\u00a0and Entringer showed in 1983 that $\\mathrm{sat}(n,2,1)=\\lceil \\tfrac{3n}2\\rceil$. The present authors proved\u00a0that for $k\\geqslant 3$ and $\\ell=1$, as well as for all $0.8k\\leqslant \\ell\\leq k-1$, $\\mathrm{sat}(n,k,\\ell)=\\Theta(n^{\\ell})$. In this paper we prove two upper bounds which cover the remaining\u00a0range of $\\ell$. The first, quite technical one, restricted to $\\ell\\geqslant\\frac{k+1}2$, implies in\u00a0particular that for $\\ell=\\tfrac23k$ and $\\ell=\\tfrac34k$ we have $\\mathrm{sat}(n,k,\\ell)=O(n^{\\ell+1})$. Our main result provides an upper bound $\\mathrm{sat}(n,k,\\ell)=O(n^{\\frac{k+\\ell}2})$ valid for all $k$ and $\\ell$.\u00a0In the smallest open case we improve it further to $\\mathrm{sat}(n,4,2)=O(n^{\\frac{14}5})$.<\/jats:p>","DOI":"10.37236\/5252","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T20:05:27Z","timestamp":1578686727000},"source":"Crossref","is-referenced-by-count":1,"title":["Upper Bounds on the Minimum Size of Hamilton Saturated Hypergraphs"],"prefix":"10.37236","volume":"23","author":[{"given":"Andrzej","family":"Ruci\u0144ski","sequence":"first","affiliation":[]},{"given":"Andrzej","family":"\u017bak","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2016,10,28]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p12\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v23i4p12\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T05:11:17Z","timestamp":1579237877000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v23i4p12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,10,28]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2016,10,14]]}},"URL":"https:\/\/doi.org\/10.37236\/5252","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2016,10,28]]},"article-number":"P4.12"}}