AI 中文总结
研究c-空心图中击中所有极大独立集的问题，通过随机构造给出τ(G)=Ω(n^(1/3)/log n)的下界，补充相关下界，同时证明该猜想对余图和分裂图类成立。
AI 中文摘要
固定常数0 < c < 1。若n个顶点的图G中每个极大独立集的大小至少为cn，则称G为c-空心图。用τ(G)表示顶点集T⊆V(G)的最小规模，使得G中的每个极大独立集都与T相交。1991年，Bollobás、Erdős和Tuza猜想若G是c-空心图，则τ(G)=o(n)。我们通过随机构造表明存在c-空心图使得τ(G)=Ω(n^(1/3)/log n)，这是对该猜想的首个非平凡下界约束。我们还表明该猜想对余图和分裂图类以强形式成立。
英文摘要
Fix a constant $c$ with $0<c<1$. We say a graph $G$ on $n$ vertices is $c$-hollow if every maximal independent set of $G$ has size at least $cn$. Denote by $τ(G)$ the size of a smallest set of vertices $T\subseteq V(G)$ such that every maximal independent set in $G$ intersects $T$, i.e., $T$ is a transversal for the family of maximal independent sets. In 1991, Bollobás, Erdős, and Tuza conjectured that if $G$ is $c$-hollow, then $τ(G)=o(n)$. Using a random construction, we show there exist $c$-hollow graphs with $τ(G)=Ω\left(\frac{n^{1/3}}{\log n }\right)$, establishing the first nontrivial lower bound constraining the conjecture and complementing a closely related lower bound due to Alon for maximum independent sets. We also show the conjecture holds in a strong form for the class of cographs and split graphs.
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