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- W1506619093 abstract "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." @default.
- W1506619093 created "2016-06-24" @default.
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- W1506619093 date "2009-09-25" @default.
- W1506619093 modified "2023-10-18" @default.
- W1506619093 title "Rainbow Matchings in $r$-Partite $r$-Graphs" @default.
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- W1506619093 doi "https://doi.org/10.37236/208" @default.
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