On the girth of voltage graph lifts

We study the potential and limitations of the voltage graph construction for producing small regular graphs of large girth. We determine the relation between the girth of the base graph and the lift, and we show that any base graph can be lifted to a graph of arbitrarily large girth. We determine upper bounds on the girths of voltage graphs with respect to the nilpotency class in the case of nilpotent groups or the length of the derived series in the case of solvable voltage groups. These results suggest the use of perfect groups, which we use to construct the smallest known cubic graphs of girths 29 and 30. We also construct the smallest known (5, 10)-graphs and (7, 8)-graphs.