On the girth of random cayley graphs

A. Gamburd; S. Hoory; M. Shahshahani; A. Shalev; B. Virág

doi:10.1002/rsa.20266

On the girth of random cayley graphs

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

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

32 Scopus citations

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|)^α with α < 1. Several conjectures and open questions are presented.

Original language	English
Pages (from-to)	100-117
Number of pages	18
Journal	Random Structures and Algorithms
Volume	35
Issue number	1
DOIs	https://doi.org/10.1002/rsa.20266
State	Published - 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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10.1002/rsa.20266

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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.",

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On the girth of random cayley graphs

Abstract

Bibliographical note

Keywords

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