Hyperfinite graph limits

TL;DR

This paper extends the concept of hyperfinite graph families to a compactification of finite bounded-degree graphs and proves that convergence to a hyperfinite limit implies the sequence's hyperfiniteness.

Contribution

It introduces a new extension of hyperfiniteness to a compactified setting and establishes that hyperfiniteness of a limit implies hyperfiniteness of converging sequences.

Findings

Hyperfiniteness can be extended to a compactification of graphs.

Sequences converging to a hyperfinite limit are themselves hyperfinite.

The work bridges graph limits and hyperfiniteness properties.

Abstract

G\'abor Elek introduced the notion of a hyperfinite graph family: a collection of graphs is hypefinite if for every $\epsilon>0$ 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 $\epsilon|V(G)|$. We presently extend this notion to a certain compactification of finite bounded-degree graphs, and show that if a sequence of finite graphs converges to a hyperfinite limit, then the sequence itself is hyperfinite.