Dirac's theorem and the switch geometry of perfect matchings

Abstract

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 $\delta(G)$ 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 $\delta(G) \le n/2-1$. 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 $\delta(G)$ of $G$? We show that if $\delta(G) \ge \lfloor2n/3\rfloor+1$, then $\mathcal H_2(G)$ must be connected and an expander, while for each $\delta\le \lfloor(2n-2)/3\rfloor$ there are $n$-vertex graphs $G$ with minimum degree $\delta$ such that $\mathcal H_2(G)$ is disconnected. We also show that, if $\delta(G) \ge n/2+2$, then $\mathcal H_3(G)$ must be connected and an expander, while for each $\delta\le n/2-C_k$ there are $n$-vertex graphs $G$ with minimum degree $\delta$ 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 $\delta(G) \ge \frac{n}2-\varepsilon kn$ 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 $\delta(G) \ge n/2+1$, then $\mathcal H_2(G)$ must have positive minimum degree. For $k\ge 3$, we show how this threshold for $\delta(G)$ is related to the notorious Caccetta-H\"aggkvist conjecture.