Approximating fractionally isomorphic graphons

TL;DR

This paper proves that fractionally isomorphic graphons can be approximated by finite graphs in the cut distance, extending fractional isomorphism concepts from graphs to graphons and enabling approximation of regular graphons by regular graphs.

Contribution

It establishes the approximation of fractionally isomorphic graphons by finite graphs, addressing a key open question and extending fractional isomorphism theory.

Findings

Fractionally isomorphic graphons can be approximated by finite graphs in the cut distance.

Every regular graphon can be approximated by regular graphs.

The main question from Grebík and Rocha (2022) is answered affirmatively.

Abstract

Greb\'ik and Rocha [Fractional Isomorphism of Graphons, Combinatorica 42, pp 365-404 (2022)] extended the well studied notion of fractional isomorphism of graphs to graphons. We prove that fractionally isomorphic graphons can be approximated in the cut distance by fractionally isomorphic finite graphs. This answers the main question from ibid. As an easy but convenient corollary, we deduce that every regular graphon can be approximated by regular graphs.