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

China’s Nuclear Forces:Operations, Training, Doctrine, Command, Control, and Campaign Planning

In China’s Nuclear Forces Larry Wortzel has delivered an exceptional mono- graph that demands the attention of both nuclear strategists and China ex- perts. The author, a leading authority on China, Asia, national security, and military strategy, is currently serving as a commissioner on the congressionally mandated U.S.-China Economic and Security Review Commission. He previ- ously served as the director of the Asian Studies Center and vice president for foreign policy at the Heritage Founda- tion. Wortzel’s distinguished thirty- two-year career in the U.S. armed forces, during which time he served as both assistant Army attaché and then attaché at the American embassy in China, culminated with an assignment as director of the Strategic Studies Insti- tute at the Army War College