On the girth of random cayley graphs

A. Gamburd*, S. Hoory, M. Shahshahani, A. Shalev, B. Virág

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

32 Scopus citations

Abstract

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

Original languageAmerican English
Pages (from-to)100-117
Number of pages18
JournalRandom Structures and Algorithms
Volume35
Issue number1
DOIs
StatePublished - Aug 2009

Bibliographical note

Funding Information:
This work was supported by the Fay Fuller Foundation, a University of Adelaide Postgraduate Scholarship (to DHP), Senior Research Fellowships (508098 and 1042589) and Project Grant (626936) from the National Health and Medical Research Council of Australia (to SMP). We thank Julia Zebol for technical assistance and Andrew Bert for assistance with Figure preparation.

Keywords

  • Cayley graphs
  • Girth
  • Random

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