三角形独立数和覆盖数的精确渐近性
Sharp asymptotics for triangle independence and covering numbers
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中文总结 AI 辅助
该研究确定了图论中三角形独立数与覆盖数之和的渐近常数，证明其极限值为$3/2$，解决了Erdős等人提出的相关渐近常数问题。
中文摘要 AI 辅助
对于图$G$，令$\u03b1_1(G)$为最多包含每个三角形中一条边的边集的最大规模，$\u03c4_1(G)$为与每个三角形都相交的边集的最小规模。Erdős、Gallai和Tuza证明，对每个含$m$条边的图，$\u03b1_1(G)+\u03c4_1(G)=\u03a9(m^{2/3})$，并询问最优渐近常数。我们证明当$m\u2192\u221e$时，所有含$m$条边的图中$\u03b1_1(G)+\u03c4_1(G)$与$m^{2/3}$比值的最小值极限为$3/2$，确定了精确常数为$3/2$并解决了该问题。
英文摘要
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.
