There is a wellknown conjecture that all connected Cayley graphs are Hamiltonian. For how large a value of n has the conjecture been verified (i.e., for all groups whose order is at most n)?
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$\begingroup$ Not an answer, but: "Almost all Cayley graphs are Hamiltonian": Jixiang, Meng, and Huang Qiongxiang. Acta Mathematica Sinica 12.2 (1996): 151155. $\endgroup$ – Joseph O'Rourke Mar 20 '14 at 23:18

$\begingroup$ I think that I did cubic Cayley graphs on up to 1000 vertices quite a few years ago, in the general hope that if any counterexample to the conjecture exists then cubic is the most likely place. $\endgroup$ – Gordon Royle Mar 21 '14 at 0:13
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According to : http://arxiv.org/pdf/1009.5795v3.pdf this is known for all n up to 120 except 72, 96, 108 and 120.

$\begingroup$ I've learned this has been checked to 1280 by Potocnik, Spiga and Verret, J. Symbolic Computation 50 (2013) 465477. $\endgroup$ – user39684 Mar 25 '14 at 19:06

$\begingroup$ It has been checked to 1280 that, in the cubic vertextransitive case, only the four wellknown exceptions occur. The question asked for Cayley graphs of arbitrary valency. $\endgroup$ – verret Mar 26 '14 at 1:43