polycirculant conjecture - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T01:04:04Z http://mathoverflow.net/feeds/question/111537 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111537/polycirculant-conjecture polycirculant conjecture majid arezoomand 2012-11-05T09:06:51Z 2012-11-05T10:43:17Z <p>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 <a href="http://arxiv.org/abs/math.GM/0204209" rel="nofollow">http://arxiv.org/abs/math.GM/0204209</a>, April 2002. 2. E. Mwambene, "A proof of the polycirculant conjecture", available on <a href="http://arxiv.org/abs/math/0506617" rel="nofollow">http://arxiv.org/abs/math/0506617</a>, 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).</p> <p>So I want to know that the polycirculat conjecture is proved or not?</p> http://mathoverflow.net/questions/111537/polycirculant-conjecture/111544#111544 Answer by Michael Giudici for polycirculant conjecture Michael Giudici 2012-11-05T10:36:37Z 2012-11-05T10:43:17Z <p>The Conjecture is still open. </p> <p>Lemma 5 of math.GM/0204209 is false. For example, any primitive group on a prime number of points is a counterexample.</p> <p>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.</p>