On the girth of random Cayley graphs

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 log d-1|G|-dim(G). For the symmetric p-groups the girth is between loglog |G| and (log |G|)α with α < 1. Several conjectures and open questions are presented. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009

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