On problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey

Fixed-point-free permutations‎, ‎also known as derangements‎, ‎have been studied for centuries‎. ‎in particular‎, ‎depending on their applications‎, ‎derangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics‎: ‎from number theory to algebraic graph theory‎. ‎substantial progress has been made on the study of derangements‎, ‎many long-standing open problems have been solved‎, ‎and many new research problems have arisen‎. ‎the results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs‎. ‎the methods used in this area range from deep group theory‎, ‎including the classification of the finite simple groups‎, ‎to combinatorial techniques‎. ‎this article is devoted to surveying results‎, ‎open problems and methods in this area‎.