Expected Maximum Block Size in Critical Random Graphs

Abstract

Abstract Let G(n,M) be a uniform random graph with n vertices and M edges. Let ${\wp_{n,m}}$ be the maximum block size of G(n,M), that is, the maximum size of its maximal 2-connected induced subgraphs. We determine the expectation of ${\wp_{n,m}}$ near the critical point M = n/2. When n − 2M ≫ n2/3, we find a constant c1 such that $$c_1 = \lim_{n \rightarrow \infty} \left({1 - \frac{2M}{n}} \right) \,\E({\wp_{n,m}}).$$ Inside the window of transition of G(n,M) with M = (n/2)(1 + λn−1/3), where λ is any real number, we find an exact analytic expression for $$c_2(\lambda) = \lim_{n \rightarrow \infty} \frac{\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}}{n^{1/3}}.$$ This study relies on the symbolic method and analytic tools from generating function theory, which enable us to describe the evolution of $n^{-1/3}\,\E{\left({\wp_{n,{{(n/2)}({1+\lambda n^{-1/3}})}}}\right)}$ as a function of λ.