{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:17Z","timestamp":1753893857396,"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>In this paper we generalize the concept of uniquely $K_r$-saturated graphs to hypergraphs. Let $K_r^{(k)}$ denote the complete $k$-uniform hypergraph on $r$ vertices. For integers $k,r,n$ such that $2\\leqslant k &lt;r&lt;n$, a $k$-uniform hypergraph $H$ with $n$ vertices is uniquely $K_r^{(k)}$-saturated\u00a0if $H$ does not contain $K_r^{(k)}$ but adding to $H$ any $k$-set that is not a hyperedge of $H$ results in exactly one\u00a0copy of $K_r^{(k)}$. Among uniquely $K_r^{(k)}$-saturated hypergraphs, the interesting ones are the primitive\u00a0ones that do not have a dominating vertex\u2014a vertex belonging to all possible ${n-1\\choose k-1}$ edges. Translating the concept to the complements of these hypergraphs, we obtain a natural restriction of $\\tau$-critical hypergraphs: a hypergraph $H$ is uniquely $\\tau$-critical\u00a0if for every edge $e$, $\\tau(H-e)=\\tau(H)-1$ and $H-e$ has a unique transversal of size $\\tau(H)-1$.We have two constructions for primitive uniquely $K_r^{(k)}$-saturated hypergraphs. One shows that for $k$ and $r$ where $4\\leqslant k&lt;r\\leqslant 2k-3$, there exists such a hypergraph for every $n&gt;r$. This is in contrast to the case $k=2$ and $r=3$ where only the Moore graphs of diameter two have this property. Our other construction keeps $n-r$ fixed; in this case we show that for any fixed $k\\ge 2$ there can only be finitely many examples. We give a range for $n$ where these hypergraphs exist. For $n-r=1$ the range is completely determined: $k+1\\leqslant n \\leqslant {(k+2)^2\\over 4}$. For larger values of $n-r$ the upper end of our range reaches approximately half of its upper bound. The lower end depends on the chromatic number of certain Johnson graphs.<\/jats:p>","DOI":"10.37236\/7534","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:00:49Z","timestamp":1578668449000},"source":"Crossref","is-referenced-by-count":0,"title":["Uniquely $K_r^{(k)}$-Saturated Hypergraphs"],"prefix":"10.37236","volume":"25","author":[{"given":"Andra\u0301s","family":"Gya\u0301rfa\u0301s","sequence":"first","affiliation":[]},{"given":"Stephen G.","family":"Hartke","sequence":"additional","affiliation":[]},{"given":"Charles","family":"Viss","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2018,11,16]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p35\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v25i4p35\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:21:28Z","timestamp":1579234888000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v25i4p35"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,11,16]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2018,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/7534","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2018,11,16]]},"article-number":"P4.35"}}