MarriageTheoremのこと

_ プレプリント確認状況：arXiv:math 3月22日分まで、IACR ePrint 2012/531まで

_ 気になった論文１：From edge-disjoint paths to independent paths (Serge Gaspers, arXiv:1203.4483)

Let f(k) denote the maximum such that every simple undirected graph containing two vertices s,t and k edge-disjoint s-t paths, also contains two vertices u,v and f(k) independent u-v paths. Here, a set of paths is independent if none of them contains an interior vertex of another. We prove that f(k)=k if k<3, and f(k)=3 otherwise.

_ 気になった論文２：Elements with finite Coxeter part in an affine Weyl group (Xuhua He, Zhongwei Yang, arXiv:1203.4680)

Let $W_a$ be an affine Weyl group and $\eta:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $\eta(w)$ is conjugate to a Coxeter element of $W_0$. The elements with finite Coxeter part is a union of conjugacy classes of $W_a$. We show that for each conjugacy class $\mathcal{O}$ of $W_a$ with finite Coxeter part there exits a unique maximal proper parabolic subgroup $W_J$ of $W_a$, such that the set of minimal length elements in $\mathcal{O}$ is exactly the set of Coxeter elements in $W_J$. Similar results hold for twisted conjugacy classes.