Forcing quasirandomness with triangles

TL;DR

This paper investigates forcing pairs in quasirandom graphs, especially involving triangles, and demonstrates how replacing edges with triangles preserves the forcing property, advancing understanding of graph quasirandomness and related conjectures.

Contribution

It proves that if a pair involving an edge and a bipartite graph is forcing, then replacing edges with triangles yields a forcing pair involving a triangle, extending known results.

Findings

$(K_3,C'_4)$ is a forcing pair.

Replacing edges with triangles preserves forcing pairs.

Strengthens previous results by Simonovits, Sós, and Conlon et al.

Abstract

We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families $\\mathcal F$ of graphs with the property that if a large graph $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in(0,1]$ for every $F\in \mathcal F$, then $G$ is quasirandom with density $p$. Such families $\mathcal F$ are said to be forcing. Several forcing families were found over the last three decades and characterising all bipartite graphs $F$ such that $(K_2,F)$ is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely related to Sidorenko's conjecture. In fact, most of the known forcing families involve bipartite graphs only. We consider forcing pairs containing the triangle $K_3$. In particular, we show that if $(K_2,F)$ is a forcing pair, then so is $(K_3,F')$, where $F'$ is obtained from $F$ by replacing every edge of $F$ by a triangle (each of which introduces a new vertex). For the proof we first show that $(K_3,C'_4)$ is a forcing pair, which strengthens related results of Simonovits and S\'os and of Conlon et al.