COMBINATORICA Bolyai Society – Springer-Verlag 0209–9683/102/$6.00 c 2002 J´ anos Bolyai Mathematical Society Combinatorica 22 (1) (2002) 1–18 RANDOM GRAPH COVERINGS I: GENERAL THEORY AND GRAPH CONNECTIVITY ALON AMIT, NATHAN LINIAL* Received June 21, 1999 Revised November 16, 2000 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 δ- connected. In subsequent papers we will address other graph properties, such as girth, expansion and chromatic number. 1. Introduction The notion of covering maps between graphs is essentially a restriction to the case of graphs (as, say, one dimensional simplicial complexes) of the general topological notion of covering map. It is described in purely combinatorial terms as a mapping of graphs that maps the neighbours of a vertex one-to- one onto the neighbours of its image vertex (We will later refine this slightly to properly account for multiple edges and loops). Covering maps have received considerable attention from several different points of view. For example, Leighton’s remarkable theorem on common Mathematics Subject Classification (2000): 05C80, 05C10, 05C40 * Work supported in part by grants from the Israel Academy of Aciences and the Bina- tional Israel-US Science Foundation.