arXiv 1306.36112013-06math.CO

完美匹配图中的测地线

Geodesics in a Graph of Perfect Matchings

Roy H. Jennings

发表信息：: Séminaire Lotharingien de Combinatoire 74 (2017): B74e

摘要：

设$\mathscr{P}_{m}$是完全图$K_{2m}$中完美匹配集上的图，其中如果两个完美匹配的对称差是一个长度为4的圈，则两个完美匹配由一条边连接。本文研究了$\mathscr{P}_{m}$中的测地线。$\mathscr{P}_{m}$的直径以及每个顶点的偏心率显示为$m-1$。给出了两个证明，证明了任意两对极间的测地线个数为$m^{m-2}$。第一个是通过递归公式的直接证明，第二个是通过减少到对称群$S_m$中给定$m$ -圈的最小分解的数目。文中还给出了$\mathscr{P}_{m}$中任意两个匹配之间的测地线数目的显式公式。设$\mathscr{M}_m$是圆上具有与$\mathscr{P}_m$中相同邻接条件的$2m$标记点的非交叉完美匹配集合上的图。$\mathscr{M}_m$是$\mathscr{P}_m$的一个诱导子图，证明了$\mathscr{M}_m$恰好有一对极之间具有最大测地线数（$m^{m-2}$）.

英文摘要：

Let $\mathscr{P}_{m}$ be the graph on the set of perfect matchings in the complete graph $K_{2m}$, where two perfect matchings are connected by an edge if their symmetric difference is a cycle of length four. This paper studies geodesics in $\mathscr{P}_{m}$. The diameter of $\mathscr{P}_{m}$, as well as the eccentricity of each vertex, are shown to be $m-1$. Two proof are given to show that the number of geodesics between any two antipodes is $m^{m-2}$. The first is a direct proof via a recursive formula, and the second is via reduction to the number of minimal factorizations of a given $m$-cycle in the symmetric group $S_m$. An explicit formula for the number of geodesics between any two matchings in $\mathscr{P}_{m}$ is also given. Let $\mathscr{M}_m$ be the graph on the set of non-crossing perfect matchings of $2m$ labeled points on a circle with the same adjacency condition as in $\mathscr{P}_m$. $\mathscr{M}_m$ is an induced subgraph of $\mathscr{P}_m$, and it is shown that $\mathscr{M}_m$ has exactly one pair of antipodes having the maximal number ($m^{m-2}$) of geodesics between them.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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