# Expected size of matchings in a cubic graph

Let $$G$$ be a random cubic graph on $$n$$ vertices. Let $$M$$ be the set of (not necessarily maximum) matchings of $$G$$. What is the expected size (i.e. number of edges) of an element of $$M$$?

In other words, what is the typical size of a matching in a typical cubic graph?

The Bethe lattice approximation gives the number $$\frac{3}{10}n$$, which coincides with experimental results. I'm wondering whether it is the true number.

This has been done before, but I can't find it. I'll outline how it can be proved, without dotting all the "$$i$$"s.
Consider the $$d$$-regular case, for $$n$$ vertices. The expected number of matchings with $$K$$ edges can be obtained by dividing two values from (for example) Thm 1 in this paper and multiplying by the number of positions that a matching can occupy. The result is a bit of a mess. However, if $$E(K)$$ is the expected number of matchings with $$K$$ edges, we find that, for constant $$c\in(0,\frac12)$$, $$\frac{E(cn+1)}{E(cn)} \to \frac{(1-2c)^2 d^2}{2 (d-2c)c}.$$ The maximum occurs when this ratio is 1, which is when $$c = \frac{(2d+1-\sqrt{4d-3})\,d}{4(d^2+1)},$$ which is indeed equal to $$\frac3{10}$$ when $$d=3$$.