FORCING QUASIRANDOMNESS WITH TRIANGLES

Abstract

We studyforcing pairsforquasirandom 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 beforcing. Several forcing families were found over the last three decades and characterizing 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$(K_{3},F^{\rhd })$, where$F^{\rhd }$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 $(K_{3},C_{4}^{\rhd })$is a forcing pair, which strengthens related results of Simonovits and Sós and of Conlon et al.