On the number of triangles in $K_4$-free graphs

Abstract

Erd\H{o}s asked whether for any $n$-vertex graph $G$, the parameter $p^*(G)=\min \sum_{i\ge 1} (|V(G_i)|-1)$ is at most $\lfloor n^2/4\rfloor$, where the minimum is taken over all edge decompositions of $G$ into edge-disjoint cliques $G_i$. In a restricted case (also conjectured independently by Erd\H{o}s), Gy\H{o}ri and Keszegh [Combinatorica, 37(6) (2017), 1113--1124] proved that $p^*(G)\leq \lfloor n^2/4\rfloor$ for all $K_4$-free graphs $G$. Motivated by their proof approach, they conjectured that for any $n$-vertex $K_4$-free graph $G$ with $e$ edges, and any greedy partition $P$ of $G$ of size $r$, the number of triangles in $G$ is at least $r(e-r(n-r))$. If true, this would imply a stronger bound on $p^*(G)$. In this paper, we disprove their conjecture by constructing infinitely many counterexamples with arbitrarily large gap. We further establish a corrected tight lower bound on the number of triangles in such graphs, which would recover the conjectured bound once some small counterexamples we identify are excluded.