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arXiv · 0707.1833

On the girth of random Cayley graphs

Alex Gamburd·Shlomo Hoory·Mehrdad Shahshahani·Aner Shalev·Balint Virag
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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|)^alpha with alpha<1. Several conjectures and open questions are presented.

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Alex Gamburd, Shlomo Hoory, Mehrdad Shahshahani, Aner Shalev, Balint Virag. 2007-07-12. On the girth of random Cayley graphs. https://doi.org/10.1002/rsa.20266

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