Fractionally Isomorphic Graphs and Graphons

Abstract

Fractional isomorphism is a well-studied relaxation of graph isomorphism with a very rich theory. Grebík and Rocha [Combinatorica 42, pp 365--404 (2022)] developed a concept of fractional isomorphism for graphons and proved that it enjoys an analogous theory. In particular, they proved that if $G_1,G_2,\ldots$ converge to a graphon $U$, $H_1,H_2,\ldots$ converge to a graphon $W$ and each $G_i$ is fractionally isomorphic to $H_i$, then $U$ is fractionally isomorphic to $W$. Answering the main question from \emph{ibid}, we prove the converse of the statement above: If $U$ and $W$ are fractionally isomorphic graphons, then there exist sequences of graphs $G_1,G_2,\ldots$ and $H_1,H_2,\ldots$ which converge to $U$ and $W$ respectively and for which each $G_i$ is fractionally isomorphic to $H_i$. As an easy but convenient corollary of our methods, we get that every regular graphon can be approximated by regular graphs.