Universality of the matching number in percolated regular graphs

TL;DR

The paper establishes a universal limit for the matching number in percolated $d$-regular graphs: for $p=c/d$, the rescaled matching number $ u((G_d)_p)/|V(G_d)|$ converges in probability to $F(c)$, defined via a fixed-point relation, regardless of the underlying regular graph sequence as $d\to\infty$. The authors prove a quantitative local convergence of $(G_d)_p$ to a Galton–Watson tree with offspring distribution $ ext{Po}(c)$, and leverage a result of Bordenave–Lelarge–Salez to transfer this local limit to the matching-number limit, yielding universality with the binomial random graph $G(n,c/n)$. This approach generalizes Karp–Sipser-type analyses from the complete graph to broad host graphs, provided the degree grows, and it connects percolation on regular graphs to classical random-graph phenomena. The findings offer a robust framework for understanding how local graph structure governs global matching properties in sparse, expanding regimes, with potential extensions to multipartite and approximately regular graphs.

Abstract

Fix a sequence of $d$-regular graphs $(G_d)_{d\in \mathbb{N}}$ and denote by $G_{d,p}$ the graph obtained from $G_d$ after edge-percolation with probability $p=c/d$, for a constant $c>0$. We prove a quantitative local convergence of $(G_{d,p})_{d\in \mathbb{N}}$. In combination with results of Bordenave, Lelarge and Salez, it implies that the rescaled matching number of $G_{d,p}$ is asymptotically equivalent to that of the binomial random graph $G(n,c/n)$.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• proof
• Theorem 3.5
• Lemma 3.6
• proof