Probability of a cubic graph containing a perfect matching Let $G_n$ be a uniformly random cubic graph on $n$ vertices (cubic=all vertices have degree 3). What's the probability $p_n$ that $G_n$ contains a perfect matching? Does it have a limit? 
Note that by Peterson's theorem a cubic graph contains a perfect matching if it is bridgeless. So a similar question would be: what's the probability $q_n$ that a cubic graph is bridgeless? 
I'm making a vague guess that $q_n\rightarrow 1$, which would imply $p_n\rightarrow 1$. 
 A: Robinson and Wormald showed in 1994 that if as $n$ tends to infinity, with probability approaching $1$ a $3$-regular graph can actually be partitioned into the union of a Hamiltonian Cycle and a perfect matching.  Their argument is quite intricate, but for the result you're aiming for, there's a somewhat shorter argument that shows the graph is almost surely $3$-connected (in particular, bridgeless), which goes roughly as follows:
Step 1: Instead of working with a uniform $3$-regular graph, instead work with a multigraph drawn from the so-called "configuration model" described in the notes linked to from your comment. This in general may not give you a graph (it might have self-loops or multiple edges), but you can show it is a simple graph with positive probability.  So if an event has probability going to $0$ in the configuration model, it also has probability going to $0$ for random $3$-regular graphs.
Step 2: Within the configuration model, take the union bound over all $(A,S)$ with $|A| \leq n/2$ and $|S| \leq 2$ of the probability removing $S$ disconnects $A$ from the rest of the graph.  For full details, see appendix $1$ of David Ellis' lecture notes here.  
