By the polycirculant conjecture, every vertex-transitive graph is a polycirculant graph (D. Marusic 1981 and D. Jordan 1988).
There are two papers that claim to prove this conjecture:
1. A. Golubchik, "On the polycirculant conjecture", available on <http://arxiv.org/abs/math.GM/0204209>, April 2002.
2. E. Mwambene, "A proof of the polycirculant conjecture", available
on <http://arxiv.org/abs/math/0506617>, Jun 2005.
But I find some papers that proved the conjecture in special cases, after 2005. For example
(a) Every vertex-transitive graph of valency four is a polycirculant (E.
Dobson et.al 2007)
(b) All vertex-transitive locally-quasiprimitive graphs have a semiregular
automorphism (M. Giudici and J. Xu 2007).
(c) Every connected distance-transitive graph admits a semiregular
automorphism (K. Kuntar and P.Sparl 2010).


So I want to know that the polycirculat conjecture is proved or not?



polycirculant conjecture
The Conjecture is still open. 


Lemma 5 of math.GM/0204209 is false. For example, any primitive group on a prime number of points is a counterexample.


Lemma 6 of math/0506617 is also false. Any transitive permutation group without a derangement of prime order satisfies the hypotheses and does not contain a semiregular element. (Any semiregular element has a power that is stil semiregular and of prime order.) Such groups exist, such as $M\_{11}$ acting on the twelve points.


