Triangles in graphs without the expansion of $4$-cycle

TL;DR

This paper resolves a conjecture about the maximum number of triangles in graphs without a specific expanded 4-cycle, confirming the unique counterexample to the conjecture.

Contribution

It completes the proof of a conjecture by resolving the remaining case involving the expanded 4-cycle, identifying it as the only counterexample.

Findings

Confirmed the conjecture for all cases except the expanded 4-cycle.

Identified the expanded 4-cycle as the only counterexample.

Resolved the remaining open case in the conjecture.

Abstract

The expansion $F^{\triangle}$ of a graph $F$ is the graph obtained from $F$ by replacing each edge with a triangle. Lv \etal proposed a conjecture on the maximum number of triangles in a graph without $P_k^{\triangle}$ or $C_k^{\triangle}$ for every $k \ge 4$. Their conjecture was confirmed in previous work for $P_k^{\triangle}$ when $k \ge 4$ and $C_k^{\triangle}$ when $k \ge 5$. In this note, we resolve the remaining case $C_4^{\triangle}$, demonstrating that this is the only counterexample to their conjecture.