On the girth of random Cayley graphs

Details On the girth of random Cayley graphs On the girth of random Cayley graphs

2007-07-12

arxiv.org/abs/0707.1833v1

Random Structures Algorithms 35 (2009), no. 1, 100-117

Mathematics

Probability

20 pages

Scientific paper

10.1002/rsa.20266

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.

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