Hyperfinite graph limits

Abstract

Gábor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every ∈ > 0 there is some finite k such that each graph G in the collection can be broken into connected components of size at most k by removing a set of edges, of size at most ∈ |V (G)|. We presently extend this notion to a certain compactification of finite boundeddegree graphs and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite. © 2008 American Institute of Mathematical Sciences.