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arXiv · 2608.15561

Sharp asymptotics for triangle independence and covering numbers

Zhen Liu·Qinghou Zeng
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Abstract

For a graph $G$, let $α_1(G)$ be the maximum size of an edge set containing at most one edge from every triangle, and let $τ_1(G)$ be the minimum size of an edge set meeting every triangle. Erdős, Gallai, and Tuza proved that $α_1(G)+τ_1(G)=Ω(m^{2/3})$ for every $m$-edge graph and asked for the optimal asymptotic constant. We prove $$\lim_{m\to\infty} \min_{G,\,|E(G)|=m} \frac{α_1(G) + τ_1(G)}{m^{2/3}} = \frac{3}{2},$$ thereby establishing that the sharp constant is $3/2$ and solving the problem.

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Zhen Liu, Qinghou Zeng. 2026-08-16. Sharp asymptotics for triangle independence and covering numbers. https://arxiv.org/abs/2608.15561

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