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

TL;DR

This paper proves a long-standing conjecture that $K_4$-free graphs with a certain number of edges contain a specific number of edge-disjoint triangles, advancing understanding of triangle packings in such graphs.

Contribution

It confirms the conjecture that $K_4$-free graphs with $n$ vertices and $loor{n^2/4}+k$ edges contain exactly $k$ edge-disjoint triangles, resolving a 25-year-old problem.

Findings

Confirmed the conjecture for all $K_4$-free graphs.

Established the relationship between edges and edge-disjoint triangles.

Extended the understanding of triangle packings in $K_4$-free graphs.

Abstract

We show the quarter of a century old conjecture that every $K_4$-free graph with $n$ vertices and $\lfloor n^2/4 \rfloor +k$ edges contains $k$ pairwise edge disjoint triangles.