Let $G$ be a graph on an even number $n$ of vertices and let ${\cal M}_G$ be the collection of perfect matchings in $G$. Dirac's theorem says that if the minimum degree $δ(G)$ of $G$ is at least $n/2$, then ${\cal M}_G$ is guaranteed to be non-empty, while this is not necessarily the case if $δ(G) \le n/2-1$. Given an integer $k\ge 2$, let $\mathcal H_k(G)$ be the reconfiguration graph formed on ${\cal M}_G$ by connecting two distinct $M_1,M_2\in {\cal M}_G$ by an edge in $\mathcal H_k(G)$ if $M_1$ can be obtained from $M_2$ by switching at most $k$ edges. Besides non-emptiness, as per Dirac's theorem, what other natural properties of $\mathcal H_k(G)$ are guaranteed based on the minimum degree $δ(G)$ of $G$? We show that if $δ(G) \ge \lfloor2n/3\rfloor+1$, then $\mathcal H_2(G)$ must be connected and an expander, while for each $δ\le \lfloor(2n-2)/3\rfloor$ there are $n$-vertex graphs $G$ with minimum degree $δ$ such that $\mathcal H_2(G)$ is disconnected. We also show that, if $δ(G) \ge n/2+2$, then $\mathcal H_3(G)$ must be connected and an expander, while for each $δ\le n/2-C_k$ there are $n$-vertex graphs $G$ with minimum degree $δ$ such that $\mathcal H_k(G)$ is disconnected, for some $C_k$ depending on $k\ge 3$. Furthermore, for every $\varepsilon >0$, there exists a $c>1$ such that for every $k\ge 2$ and every large enough $n$, there are $n$-vertex graphs $G$ with $δ(G) \ge \frac{n}2-\varepsilon kn$ such that $\mathcal H_k(G)$ has at least $c^n$ components. With respect to guaranteeing that $\mathcal H_k(G)$ has positive minimum degree (or, equivalently, no isolated vertices) we show that if $δ(G) \ge n/2+1$, then $\mathcal H_2(G)$ must have positive minimum degree. For $k\ge 3$, we show how this threshold for $δ(G)$ is related to the notorious Caccetta-Häggkvist conjecture.

翻译：设$G$为具有偶数个$n$个顶点的图，${\cal M}_G$为$G$中完美匹配的集合。狄拉克定理指出：若$G$的最小度$\delta(G) \ge n/2$，则${\cal M}_G$必然非空；而当$\delta(G) \le n/2-1$时则未必成立。给定整数$k\ge 2$，定义重构图$\mathcal H_k(G)$，其顶点集为${\cal M}_G$，且当两个不同完美匹配$M_1,M_2\in {\cal M}_G$可通过至多$k$条边的切换相互转换时，在$\mathcal H_k(G)$中连接一条边。除狄拉克定理所述的非空性外，基于$G$的最小度$\delta(G)$，$\mathcal H_k(G)$还能保证哪些其他自然性质？我们证明：若$\delta(G) \ge \lfloor 2n/3\rfloor+1$，则$\mathcal H_2(G)$必然是连通的且为扩张图；而对每个$\delta \le \lfloor (2n-2)/3\rfloor$，存在具有最小度$\delta$的$n$顶点图$G$使得$\mathcal H_2(G)$不连通。进一步地，当$\delta(G) \ge n/2+2$时，$\mathcal H_3(G)$必然是连通的且为扩张图；而对每个$\delta \le n/2-C_k$（$C_k$为依赖于$k\ge 3$的常数），存在具有最小度$\delta$的$n$顶点图$G$使得$\mathcal H_k(G)$不连通。此外，对任意$\varepsilon>0$，存在常数$c>1$使得：对任意$k\ge 2$和充分大的$n$，存在满足$\delta(G) \ge n/2-\varepsilon kn$的$n$顶点图$G$，其$\mathcal H_k(G)$至少包含$c^n$个连通分支。关于保证$\mathcal H_k(G)$具有正的最小度（即无孤立顶点）的问题，我们证明：若$\delta(G) \ge n/2+1$，则$\mathcal H_2(G)$必然具有正的最小度。对$k\ge 3$，我们展示了该$\delta(G)$阈值与著名的卡塞塔-海格奎斯特猜想之间的关系。