I am asking the question on a slightly abstract level and it may depend on the specifics but it would be great to have related references or ideas.
Consider the random graph model $G_{n,p}$ where its a random graph on n vertices and each edge is selected with probability $p$. I want to prove that a certain property $P$ is satisfied by graphs in this model with high probability ($\rightarrow 1$ as $n \rightarrow \infty$). I know/can prove the following two facts
- Conditioned on the event that the graph obtained from the sample is regular, the probability of the property being satisfied is very high (something like $1 - 1/n^c$)
- Given any graph G and any other graph G' that can be obtained by removing edges from G. If G satisfies the property then so does G'.
Can the above two statements be enough to make a general statement about the probability of the satisfaction of the property in $G_{n,p}$
Any references would be very helpful?