Random Graph Coverings I: General Theory and Graph Connectivity

In this paper we describe a simple model for random graphs that have an n-fold covering map onto a fixed finite base graph. Roughly, given a base graph G and an integer n, we form a random graph by replacing each vertex of G by a set of n vertices, and joining these sets by random matchings whenever the corresponding vertices are adjacent in G. The resulting graph covers the original graph in the sense that the two are locally isomorphic. We suggest possible applications of the model, such as constructing graphs with extremal properties in a more controlled fashion than offered by the standard random models, and also “randomizing” given graphs. The main specific result that we prove here (Theorem 1) is that if δ ≥ 3 is the smallest vertex degree in G, then almost all n-covers of G are δconnected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number.