{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:51Z","timestamp":1753893831395,"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>A famous conjecture (usually called Ryser's conjecture) that appeared in the PhD thesis of his student, J. R. Henderson, states that for an $r$-uniform $r$-partite hypergraph $\\mathcal{H}$, the inequality $\\tau(\\mathcal{H})\\le(r-1)\\!\\cdot\\! \\nu(\\mathcal{H})$ always holds. This conjecture is widely open, except in the case of $r=2$, when it is equivalent to K\u0151nig's theorem, and in the case of $r=3$, which was proved by Aharoni in 2001.Here we study some special cases of Ryser's conjecture. First of all, the most studied special case is when $\\mathcal{H}$ is intersecting. Even for this special case, not too much is known: this conjecture is proved only for $r\\le 5$ by Gy\u00e1rf\u00e1s and Tuza. For $r&gt;5$ it is also widely open.Generalizing the conjecture for intersecting hypergraphs, we conjecture the following. If an $r$-uniform $r$-partite hypergraph $\\mathcal{H}$ is $t$-intersecting (i.e., every two hyperedges meet in at least $t&lt;r$ vertices), then $\\tau(\\mathcal{H})\\le r-t$. We prove this conjecture for the case $t&gt; r\/4$.Gy\u00e1rf\u00e1s showed that Ryser's conjecture for intersecting hypergraphs is equivalent to saying that the vertices of an $r$-edge-colored complete graph can be covered by $r-1$ monochromatic components.Motivated by this formulation, we examine what fraction of the vertices can be covered by $r-1$ monochromatic components of different colors in an $r$-edge-colored complete graph. We prove a sharp bound for this problem.Finally we prove Ryser's conjecture for the very special case when the maximum degree of the hypergraph is two.<\/jats:p>","DOI":"10.37236\/6448","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:42:45Z","timestamp":1578670965000},"source":"Crossref","is-referenced-by-count":2,"title":["On Ryser\u2019s Conjecture for $t$-Intersecting and Degree-Bounded Hypergraphs"],"prefix":"10.37236","volume":"24","author":[{"given":"Zolt\u00e1n","family":"Kir\u00e1ly","sequence":"first","affiliation":[]},{"given":"Lilla","family":"T\u00f3thm\u00e9r\u00e9sz","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,12,22]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p40\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i4p40\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:41:53Z","timestamp":1579236113000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i4p40"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,22]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2017,10,5]]}},"URL":"https:\/\/doi.org\/10.37236\/6448","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,12,22]]},"article-number":"P4.40"}}