Dense halves in balanced 2-partition of K4-free graphs

TL;DR

This paper investigates balanced bipartitions in $K_4$-free graphs, disproves a conjecture by providing counterexamples, and establishes a new upper bound on the maximum edges in such partitions.

Contribution

The paper presents counterexamples to a conjecture on balanced 2-partitions in $K_4$-free graphs and introduces a tighter upper bound for large graphs.

Findings

Counterexamples to the conjecture are constructed.

A new upper bound of 0.074n^2 edges is proven for large graphs.

Disproves the previously believed universal bound of n^2/16.

Abstract

A balanced 2-partition of a graph is a bipartition $A,A^c$ of $V(G)$ such that $|A|=|A^c|$. Balogh, Clemen, and Lidick\'y conjectured that for every $K_4$-free graph on $n$ (even) vertices, there exists a balanced 2-partition $A,A^c$ such that $\max\{e(A),e(A^c)\}\leq n^2/16$ edges. In this paper, we present a family of counterexamples to the conjecture and provide a new upper bound ($0.074n^2$) for every sufficiently large even integer $n$.