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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?

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It is a general policy on MO not to ask about correctness of preprints claiming to prove hard conjectures. The paper math.GM/0204209 is almost surely false for it also claims a simple proof of the Feit-Thompson theorem (in 32 pages!). The paper math/0506617 does not appear to have been published in seven years. But if you want to know if the conjecture is proved or not is I suppose a reasonable, on-topic question. –  David Roberts Nov 5 '12 at 9:19
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David, I didn't know about this MO policy. I've seen a few questions on MO concerning arXiv preprints and they've often received a very large amount of interest. So long as these questions are asked in a respectful fashion, then it seems to me that MO is a reasonable place for them. (However if there's an official MO policy against such questions, then of course we should abide by that.) –  Nick Gill Nov 5 '12 at 11:24
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I think David Roberts is referring to various discussions on meta. See tea.mathoverflow.net/discussion/1422/… and the references therein, and also tea.mathoverflow.net/discussion/1447/3/… . –  HJRW Nov 5 '12 at 12:48
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1 Answer

up vote 12 down vote accepted

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.

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Many thanks for your exact answer. –  majid arezoomand Nov 5 '12 at 15:14
    
Can you give me more counterexamples to show that the lemma 5 of math. GM/0204209 is false? –  majid arezoomand Nov 5 '12 at 19:03
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How many counterexamples does it take to prove that a lemma is false? –  Gerry Myerson Nov 5 '12 at 22:27
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$A_5$ acting on 10 points is another counterexample. –  Michael Giudici Nov 6 '12 at 6:04
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Another counterexample: $M_{22}$ on 22 points is a counterexample to Lemma 5 of math.GM/0204209 . (As Gerry Myerson noted, this is overkill. However, I'm interested in the sporadic simple groups and this is a way to use one of them.) –  DavidLHarden Nov 10 '12 at 2:20
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