Group Marriage Problem

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 $\mathcal{V}$

Gallai-Edmonds Structure Theorem for Weighted Matching Polynomial

In this paper, we prove the Gallai-Edmonds structure theorem for the most general matching polynomial. Our result implies the Parter-Wiener theorem and its recent generalization about the existence of principal submatrices of a Hermitian matrix whose graph is a tree. keywords:Comment: 34 pages, 5 figure

The covering radius problem for sets of perfect matchings

Consider the family of all perfect matchings of the complete graph $K_{2n}$ with $2n$ vertices. Given any collection $\mathcal M$ of perfect matchings of size $s$, there exists a maximum number $f(n,x)$ such that if $s\leq f(n,x)$, then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. We use probabilistic arguments to give several lower bounds for $f(n,x)$. We also apply the Lov\'asz local lemma to find a function $g(n,x)$ such that if each edge appears at most $g(n, x)$ times then there exists a perfect matching that agrees with each perfect matching in $\mathcal M$ in at most $x-1$ edges. This is an analogue of an extremal result vis-\'a-vis the covering radius of sets of permutations, which was studied by Cameron and Wanless (cf. \cite{cameron}), and Keevash and Ku (cf. \cite{ku}). We also conclude with a conjecture of a more general problem in hypergraph matchings.Comment: 10 page