Let $G(n,p)$ be the usual random graph on $n$ vertices with each edge existing independently with probability $p$ (no self loops , or double edges not are allowed). I would like to simulate the distribution of a random graph given the event $T\geq a$ where $T$ is the number of triangles in the random graph. The natural approach is Metropolis Hastings. I've already found some semi-efficient algorithms that approximate the number of triangles in a given random graph, however I am still at a loss of what Markov chain to pick for a good rate of convergence. I would immensely appreciate a push in the right direction. In particular, some references would be fantastic. Thanks!