Constructions with many independent exact r-covers outside common coverings

Construct or determine the existence of graphs having many independent exact r-covers that do not arise from common-covering constructions.

Background

The paper’s general construction obtains a graph with independent exact r-covers by lifting covers from several smaller graphs through a common covering. It remains unresolved whether graphs with many such covers can be produced by fundamentally different constructions, which could potentially improve the asymptotic upper bound for the minimum order N(d).