δ-Connectivity in Random Lifts of Graphs

Amit and Linial have shown that a random lift of a connected graph with minimum degree δ > 3 is asymptotically almost surely (a.a.s.) δ-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random n-lift of a graph and a randomly generated subgroup of the symmetric group on n-elements, we show that this probability is at least 1 − O ( 1 nγ(δ) ) where γ(δ) > 0 for δ > 5 and it is strictly increasing with δ. We extend this to show that one may allow δ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. δ-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.