The number of perfect matchings, and the nesting properties, of random regular graphs

Abstract

We prove that the number of perfect matchings in G(n, d) is asymptotically normal when n is even, d → ∞ as n → ∞, and d = O(n 1/7 / log 2 n). This is the first distributional result of spanning subgraphs of G(n, d) when d → ∞.Moreover, we prove that G(n, d − 1) and G(n, d) can be coupled so that G(n,and d ≤ d ≤ n − 1 then G(n, d) and G(n, d ) can be coupled so that asymptotically almost surely (a.a.s.) G(n, d) is a subgraph of G(n, d ).