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

A balanced 2-partition of a graph is a bipartition $A,A^c$ of $V(G)$ such that $\mathrm{\mid }A\mathrm{\mid }=\mathrm{\mid }{A}^c\mathrm{\mid }$. Balogh, Clemen, and Lidický conjectured that for every $K_4$-free graph on $n$ (even) vertices, there exists a balanced 2-partition $A,A^c$ such that $\max \{e(A),e(A^c)\}\le n^2\mathrm{/}16$ 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 $n$.