I'm trying to teach myself a little more on threshold probabilities for random graphs, and I'm looking at the probability that graphs have an isolated vertex, when we add on a few restrictions (when by a 'random graph' I mean we take the set of vertices and add each possible (non-directed) edge between vertices with probability p). For example, in the standard ('unrestricted') graph on n vertices, we have something like p = log(n)/n as a probability above which we expect to get no isolated vertices a.e., and below which a.e. we get an isolated vertex. This case is well documented - however, I can find little to nothing in books or online in the case of specific types of random graph which are, for example,
k-connected/k-edge-connected bipartite/tripartite etc.
The case I'm most interested in (at the moment) is bipartite graphs, and I expect that's the next easiest case to understand too, but I can't find documentation anywhere. Is there a simple way to extend the result from normal graphs to bipartite graphs, assuming both vertex sets have the same size? I suppose my concern is that you're obviously looking at a different set of feasible graphs to the general case on 2n vertices, both fewer graphs with an isolated vertex and fewer graphs in total, so it isn't clear to me that we can immediately say the 'proportion' of graphs which have an isolated vertex will necessarily behave the same for large n.
As I mentioned above, I'd be happy to read up on anything anyone could suggest in more restricted cases, I just haven't been able to find it myself, so recommendations will be much appreciated.
Thanks very much!