AI 中文总结
本文针对Tuza猜想，通过研究二染色三角形族的关键性质，将其上界从66/22 ν(G)改进为63/22 ν(G)。
AI 中文摘要
Tuza猜想指出，对于任意图G，三角形横截集的最小规模τ(G)至多为边不交三角形集合最大规模ν(G)的两倍。本文证明τ(G)≤63/22 ν(G)，改进了Haxell于1999年给出的τ(G)≤66/23 ν(G)的已有上界。关键发现是，对于“二染色”三角形族F（每个三角形含两条蓝色边和一条红色边），可得出τ(F)≤(1+√3)ν(F)。
英文摘要
Tuza's conjecture states that for any graph $G$, the minimum size of a triangle transversal $τ(G)$ is at most twice the maximum size of a set of edge-disjoint triangles $ν(G)$. In this note, we prove $τ(G) \leq \frac{63}{22}ν(G)$, improving the previous bound $τ(G) \leq \frac{66}{23}ν(G)$ established by Haxell in 1999. The key observation is that for a "2-colorable" family of triangles $\mathcal{F}$, where each triangle has two blue edges and one red edge, we can obtain $τ(\mathcal{F})\leq (1+\sqrt{3})ν(\mathcal{F})$.
Comments4 pages
