Dirac's theorem and the switch geometry of perfect matchings

TL;DR

This paper explores how the minimum degree of a graph influences the connectivity and expansion properties of the reconfiguration graph of its perfect matchings, extending Dirac's theorem to these properties.

Contribution

It establishes new minimum degree thresholds that guarantee connectivity and expansion in the reconfiguration graph of perfect matchings, and relates these thresholds to longstanding conjectures.

Findings

If δ(G) ≥ ⌊2n/3⌋+1, then the reconfiguration graph H₂(G) is connected and an expander.

For δ(G) ≤ ⌊(2n−2)/3⌋, there exist graphs with disconnected H₂(G).

For δ(G) ≥ n/2+2, H₃(G) is connected and an expander.

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.