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arXiv · 2412.13485

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

Yue Xu·Xiao-Dong Zhang
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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 Lidický conjectured that for every $K_4$-free graph on $n$ (even) vertices, there exists a balanced 2-partition $A,A^c$ such that $\max\{e(A),e(A^c)\}\leq n^2/16$ 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 $n$.

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Yue Xu, Xiao-Dong Zhang. 2024-12-18. Dense halves in balanced 2-partition of K4-free graphs. https://arxiv.org/abs/2412.13485

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