On the girth of random Cayley graphs

TL;DR

This paper investigates the girth of random Cayley graphs across different groups, establishing lower bounds that relate girth to group size and structure, and discusses related conjectures and open problems.

Contribution

It provides new asymptotic lower bounds on the girth of random Cayley graphs for symmetric groups and algebraic groups, advancing understanding of their combinatorial properties.

Findings

Girth of symmetric group Cayley graphs is at least (log_{d-1}|G|)^{1/2}/2

Girth of algebraic group Cayley graphs is at least log_{d-1}|G|/dim(G)

Girth bounds for p-groups range between log log |G| and (log|G|)^alpha

Abstract

We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (log_{d-1}|G|)^{1/2}/2 and that random d-regular Cayley graphs of simple algebraic groups over F_q asymptotically almost surely have girth at least log_{d-1}|G|/dim(G). For the symmetric p-groups the girth is between log log |G| and (log|G|)^alpha with alpha<1. Several conjectures and open questions are presented.