Tuza猜想低于2.8的界
A Bound Below 2.8 for Tuza's Conjecture
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中文总结 AI 辅助
本文改进了Tuza猜想的界，通过红蓝着色和交换论证证明$\tau(G)\le (165/59)\nu(G)$，优于Haxell的$66/23$。
中文摘要 AI 辅助
设$\nu(G)$为图$G$中边不相交三角形的最大数量，$\tau(G)$为与每个三角形相交的最小边数。Tuza猜想$\tau(G)\le 2\nu(G)$。我们证明$\tau(G)\le (165/59)\nu(G)$。常数$165/59\approx 2.797$改进了Haxell在1999年证明的界$66/23\approx 2.870$。关键观察是，对于合适的红蓝着色，Haxell构造中剩余的族包含每个恰好有一条红边的三角形。这样的族$\mathcal{F}$允许一种交换，一旦最大打包的蓝边被删除，该交换迫使某些红边位于单个三角形中，这给出$\tau(\mathcal{F})\le (8/3)\nu(\mathcal{F})$。
英文摘要
Let $ν(G)$ be the maximum number of edge-disjoint triangles in a graph $G$ and $τ(G)$ the minimum number of edges meeting every triangle. Tuza conjectured that $τ(G)\le 2ν(G)$. We prove that $τ(G)\le (165/59)ν(G)$. The constant $165/59\approx 2.797$ improves the bound $66/23\approx 2.870$ that Haxell proved in 1999. The key observation is that, for a suitable red-blue coloring, the families left over in Haxell's construction contain every triangle with exactly one red edge. Such a family $\mathcal{F}$ admits an exchange that forces certain red edges to lie in a single triangle once the blue edges of a maximum packing are deleted, which gives $τ(\mathcal{F})\le (8/3)ν(\mathcal{F})$.
发表机构
- Shenzhen MSU-BIT University（深圳莫斯科国立技术大学）
机构由 AI 辅助整理，请以论文原文为准。
