Random induced subgraphs of Cayley graphs induced by transpositions

TL;DR

This paper investigates the structure of random induced subgraphs of Cayley graphs generated by transpositions, revealing conditions under which a giant component emerges with high probability.

Contribution

It establishes the emergence of a unique giant component in these subgraphs for certain probabilities, extending understanding of phase transitions in Cayley graph substructures.

Findings

Existence of a giant component with high probability under specified conditions.

Characterization of the giant component size using a branching process survival probability.

Conditions on the probability parameter for the emergence of the giant component.

Abstract

In this paper we study random induced subgraphs of Cayley graphs of the symmetric group induced by an arbitrary minimal generating set of transpositions. A random induced subgraph of this Cayley graph is obtained by selecting permutations with independent probability, $\lambda_n$. Our main result is that for any minimal generating set of transpositions, for probabilities $\lambda_n=\frac{1+\epsilon_n}{n-1}$ where $n^{-{1/3}+\delta}\le \epsilon_n<1$ and $\delta>0$, a random induced subgraph has a.s. a unique largest component of size $\wp(\epsilon_n)\frac{1+\epsilon_n}{n-1}n!$, where $\wp(\epsilon_n)$ is the survival probability of a specific branching process.