Dirac's theorem and the switch geometry of perfect matchings

2026-04-20

AI Summary

This study investigates how minimum degree conditions in a graph $G$ guarantee the connectivity and expansion properties of its perfect matching reconfiguration graph under at most $k$ edge-switch operations. By combining tools from extremal graph theory, explicit constructions of reconfiguration graphs, and carefully designed counterexamples, the work establishes sharp thresholds linking the minimum degree $\delta(G)$ to structural properties of the reconfiguration graphs $\mathcal{H}_2(G)$ and $\mathcal{H}_3(G)$. Specifically, it is shown that $\mathcal{H}_2(G)$ is connected and an expander whenever $\delta(G) \geq \lfloor 2n/3 \rfloor + 1$, and $\mathcal{H}_3(G)$ enjoys the same properties when $\delta(G) \geq n/2 + 2$. Below these thresholds, counterexamples with exponentially many connected components can be constructed, and for $k \geq 3$, a deep connection emerges with the Caccetta–Häggkvist conjecture.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

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 $δ(G)$ 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.

Problem

• perfect matchings
• reconfiguration graph
• minimum degree
• connectivity
• Dirac's theorem

Innovation

• perfect matchings
• reconfiguration graph
• minimum degree
• expander
• Dirac's theorem