在稠密且高度连通的图中击中最大独立集
Hitting Maximum Independent Sets in Dense and Highly Connected Graphs
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中文总结 AI 辅助
本文针对Bollobás–Erdős–Tuza猜想，通过约化原理和尖锐估计，证明了稠密高连通图中击中最大独立集的小界，并给出临界族说明方法局限。
中文摘要 AI 辅助
对于图$G$，设$h(G)$为与$G$的每个最大独立集相交的顶点集的最小基数。我们为Bollobás–Erdős–Tuza猜想建立了两个互补的约化原理：该猜想对任意图成立等价于其对任意固定正线性度的正则图成立，并且在每个遗传图类内，一个一致的次线性界等价于对每个固定正线性顶点连通度的图的次线性界。我们证明了尖锐的一般估计\\[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \\] 当分母为正时，且对平衡完全多部图取等号。因此，每个阶为$n$且满足$\kappa(G)\ge\rho n$和$\rho>1/3$的$3$-可着色图都有一个大小至多$\lfloor(\rho-1/3)^{-1}\rfloor$的击中集；直接使用$3$-着色可将此改进为当$\kappa(G)>4n/9$时大小为$6$，当$\kappa(G)>n/2$时改进为尖锐界$3$。对于独立比大于$1/4$的稠密正则图，我们获得了一个对数界，而具有线性度和线性独立数的构造表明$h(G)=\Omega(\sqrt n)$仍可能发生。我们还证明了近正则$3$-可着色图的对数界，并展示了在连通度$n/3$处的一个临界族，这解释了度盈余和度比率方法的局限性。
英文摘要
For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollobás--Erdős--Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2α(G)+δ(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $κ(G)\geρn$ and $ρ>1/3$ has a hitting set of size at most $\lfloor(ρ-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $κ(G)>4n/9$ and to the sharp bound $3$ when $κ(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=Ω(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.
