{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,29]],"date-time":"2026-09-29T18:12:40Z","timestamp":1790705560823,"version":"4.1.0"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Given a collection of  matchings ${\\cal M} = (M_1, M_2, \\ldots, M_q)$ (repetitions allowed), a matching $M$ contained in $\\bigcup {\\cal M}$ is said to be $s$-rainbow for ${\\cal M}$ if it contains representatives from $s$ matchings $M_i$ (where each edge is allowed to represent just one $M_i$). Formally, this means that there is a function $\\phi: M \\to [q]$ such that $e \\in M_{\\phi(e)}$ for all $e \\in M$, and $|Im(\\phi)|\\ge s$. Let $f(r,s,t)$ be the maximal $k$ for which there exists a set of $k$ matchings of size $t$ in some $r$-partite hypergraph, such that there is no $s$-rainbow matching of size $t$. We prove that $f(r,s,t)\\ge 2^{r-1}(s-1)$, make the conjecture that equality holds for all values of $r,s$ and $t$ and prove the conjecture when $r=2$ or $s=t=2$. In the case $r=3$, a stronger conjecture is that in a $3$-partite $3$-graph if all vertex degrees in one side (say $V_1$) are strictly larger than all vertex degrees in the other two sides, then there exists a matching of $V_1$.  This conjecture is at the same time also a strengthening of a famous conjecture, described below, of Ryser, Brualdi and Stein.  We prove a weaker version, in which the degrees in $V_1$ are at least twice as large as the degrees in the other sides. We also formulate a related conjecture on edge colorings of $3$-partite $3$-graphs and prove a similarly weakened version.<\/jats:p>","DOI":"10.37236\/208","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:24:56Z","timestamp":1578716696000},"source":"Crossref","is-referenced-by-count":47,"title":["Rainbow Matchings in $r$-Partite $r$-Graphs"],"prefix":"10.37236","volume":"16","author":[{"given":"Ron","family":"Aharoni","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eli","family":"Berger","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2009,9,25]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r119\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r119\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:40:38Z","timestamp":1579315238000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r119"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,9,25]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/208","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,9,25]]},"article-number":"R119"}}