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.
Bibliographical noteFunding 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.
- Cayley graphs