The Maximum Block Size of Critical Random Graphs

TL;DR

This paper analyzes the maximum block size in critical random graphs near the phase transition, providing exact asymptotic expectations using generating function techniques.

Contribution

It offers the first precise asymptotic formulas for the maximum block size in critical Erdős–Rényi graphs, especially within the transition window.

Findings

Expected maximum block size scales as n^{1/3} in the transition window.

Derived explicit formulas for the limit functions c_1 and c_2(λ).

Applied symbolic and analytic combinatorics methods to random graph analysis.

Abstract

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