Group Marriage Problem

Details Group Marriage Problem Group Marriage Problem

2009-12-22

arxiv.org/abs/0912.4443v1

Mathematics

Combinatorics

Scientific paper

Let $G$ be a permutation group acting on $[n]=\{1, ..., n\}$ and $\mathcal{V}=\{V_{i}: i=1, ..., n\}$ be a system of $n$ subsets of $[n]$. When is there an element $g \in G$ so that $g(i) \in V_{i}$ for each $i \in [n]$? If such $g$ exists, we say that $G$ has a $G$-marriage subject to $\mathcal{V}$. An obvious necessary condition is the {\it orbit condition}: for any $\emptyset \not = Y \subseteq [n]$, $\bigcup_{y \in Y} V_{y} \supseteq Y^{g}=\{g(y): y \in Y \}$ for some $g \in G$. Keevash (J. Combin. Theory Ser. A 111(2005), 289--309) observed that the orbit condition is sufficient when $G$ is the symmetric group $\Sym([n])$; this is in fact equivalent to the celebrated Hall's Marriage Theorem. We prove that the orbit condition is sufficient if and only if $G$ is a direct product of symmetric groups. We extend the notion of orbit condition to that of $k$-orbit condition and prove that if $G$ is the alternating group $\Alt([n])$ or the cyclic group $C_{n}$ where $n \ge 4$, then $G$ satisfies the $(n-1)$-orbit condition subject to $\V$ if and only if $G$ has a $G$-marriage subject to $\mathcal{V}$.

Affiliated with

Also associated with

No associations

Rating

Group Marriage Problem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Group Marriage Problem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Group Marriage Problem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-307542