Sharp asymptotics for triangle independence and covering numbers

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.

Disclosure

“Declaration on the Use of Generative AI The authors used ChatGPT 5.6 Pro to assist in discussing proof strategies, checking proofs, and improving exposition. References [1] Cs. Bujtás, A. Davoodi, L. Ding, E. Győri, Zs. Tuza, and D. Yang, Covering the edges of a graph with triangles, Discrete Math”

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Classification

Proof ideas or individual proof-step assistance

Multiplier

8

Verified

2026-08-23T05:17:21.169034Z

Structural counts