Transactions on Combinatorics | 2019
On problems concerning fixed-point-free permutations and on the polycirculant conjecture-a survey
Abstract
Fixed-point-free permutations\u200e, \u200ealso known as derangements\u200e, \u200ehave been studied for centuries\u200e. \u200eIn particular\u200e, \u200edepending on their applications\u200e, \u200ederangements of prime-power order and of prime order have always played a crucial role in a variety of different branches of mathematics\u200e: \u200efrom number theory to algebraic graph theory\u200e. \u200eSubstantial progress has been made on the study of derangements\u200e, \u200emany long-standing open problems have been solved\u200e, \u200eand many new research problems have arisen\u200e. \u200eThe results obtained and the methods developed in this area have also effectively been used to solve other problems regarding finite vertex-transitive graphs\u200e. \u200eThe methods used in this area range from deep group theory\u200e, \u200eincluding the classification of the finite simple groups\u200e, \u200eto combinatorial techniques\u200e. \u200eThis article is devoted to surveying results\u200e, \u200eopen problems and methods in this area\u200e.