阿尔伯森-伯曼猜想的反例：最小阶、连通性与改进的比率界
Counterexamples to the Albertson-Berman conjecture: minimum order, connectivity and an improved ratio bound
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中文总结 AI 辅助
该文针对阿尔伯森-伯曼猜想，确定其反例最小阶为29，构造了高连通度反例并改进了诱导森林的比率上界，还得到了查佩尔等猜想的反例。
中文摘要 AI 辅助
1979年，阿尔伯森（Albertson）和伯曼（Berman）提出猜想：每个平面图G都包含一个阶数至少为|V(G)|/2的诱导森林。这一长期存在的猜想最近被多个明确的反例所否定，这些反例自然引出了若干极值与结构问题，本文对此作出解答。我们结合数学论证与穷尽计算，证明反例的最小阶为29；还构造了无限多个4-连通、5-边连通的反例（并证明此类最小阶反例的阶数为41），而此前已知的所有反例的顶点连通度至多为3。此外，我们构造了一个含n个顶点的无限平面图族，其最大诱导森林的阶数至多为25/52 n，从而改进了此前的最优上界；该族还对查佩尔（Chappell）和佩尔斯马耶（Pelsmajer）关于最大度至多为d的诱导森林的猜想，产生了无限多个反例（对每个整数d≥7均成立）。
英文摘要
In 1979, Albertson and Berman conjectured that every planar graph $G$ contains an induced forest of order at least $|V(G)|/2$. This long-standing conjecture was recently disproved by several explicit counterexamples, which naturally led to several extremal and structural questions that we answer. We combine mathematical arguments and exhaustive computations to show that the minimum order of a counterexample is $29$. We also construct infinitely many $4$-connected $5$-edge-connected counterexamples (and show that the unique such counterexample of minimum order has order $41$), whereas previously all known counterexamples had vertex-connectivity at most $3$. Furthermore, we construct an infinite family of planar graphs on $n$ vertices whose maximum induced forests have order at most $\frac{25}{52}n$, thereby improving the previous best upper bound. This family also yields infinitely many counterexamples (for every integer $d \geq 7$) to a conjecture of Chappell and Pelsmajer concerning induced forests of maximum degree at most $d$.
