Balanced bipartite distance of $K_4$-free graphs

TL;DR

This paper proves that any $K_4$-free graph can be transformed into a balanced bipartite graph by removing at most $rac{n^2}{9}$ edges, confirming a conjecture and extending previous results.

Contribution

It establishes a new upper bound on edge removals needed to make $K_4$-free graphs bipartite, confirming a conjecture and generalizing prior findings.

Findings

Every $K_4$-free graph on $n$ vertices can be made bipartite by removing at most $rac{n^2}{9}$ edges.

The result confirms a conjecture of Balogh, Clemen, and Lidický.

It generalizes earlier results by Sudakov and Reiher.

Abstract

We show that every $K_4$-free graph on $n$ vertices can be made balanced bipartite by removing at most $\frac{n^2}{9}$ edges. This proves a conjecture of Balogh, Clemen, and Lidick\'{y}, and generalizes both Sudakov's result on the bipartite distance of $K_4$-free graphs and Reiher's result on the sparse half of $K_4$-free graphs.