Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionSearch and Optimization: Combinatorial Optimization
Dirac's theorem and the switch geometry of perfect matchings
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
perfect matchings
reconfiguration graph
minimum degree
connectivity
Dirac's theorem
Innovation
Methods, ideas, or system contributions that make the work stand out.
perfect matchings
reconfiguration graph
minimum degree
expander
Dirac's theorem
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