Forcing Graphs to be Forcing

TL;DR

This paper advances the understanding of Sidorenko's and the forcing conjecture by proving new classes of bipartite graphs are forcing, using algebraic methods based on Razborov's flag algebra framework.

Contribution

It extends the family of bipartite graphs known to satisfy the forcing conjecture, including certain blow-ups, subdivisions, and the box product with an edge, using algebraic techniques.

Findings

Balanced blow-ups of Sidorenko graphs are forcing.

Subdivisions of Sidorenko graphs by a forcing graph are forcing.

Cubes are shown to be forcing graphs.

Abstract

Sidorenko's conjecture states that the number of copies of any given bipartite graph in another graph of given density is asymptotically minimized by a random graph. The forcing conjecture further strengthens this, claiming that any minimizer in fact needs to be quasi-random. Here we extend the family of bipartite graphs for which the forcing conjecture is known to hold to include balanced blow-ups of Sidorenko graphs and subdivisions of Sidorenko graphs by a forcing graph. This partially generalizes results by Conlon et al. (2018) and Conlon and Lee (2021). We also show that the box product of a Sidorenko graph with an edge is forcing, partially generalizing results of Kim, Lee, and Lee (2016) and, in particular, showing that cubes are forcing. We achieve these results through algebraic arguments building on Razborov's flag algebra framework (2007). This approach additionally allows us to construct Sidorenko hypergraphs from known 2-uniform Sidorenko graphs and to study forcing pairs.