关于平面图中诱导森林的Albertson-Berman猜想的一个反例
A counterexample to the Albertson-Berman conjecture about induced forests in planar graphs
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中文总结 AI 辅助
本文独立于AI构造出Albertson-Berman猜想的反例，给出含39个顶点、$a(G)=19$的反例，还提及顶点数更多但比值略小的构造变体，否定了该猜想。
中文摘要 AI 辅助
对于图$G$，记$a(G)$为$G$中最大诱导森林的顶点数。自1979年起悬而未决的Albertson-Berman猜想指出，每个含$n$个顶点的简单平面图$G$都满足$a(G) \geq \frac{n}{2}$。尽管该猜想近期借助AI构造反例被否定，我们仍独立于AI找到该猜想的一个反例并在本文中呈现。此反例含39个顶点，其$a(G)=19$。最后，我们未给出完整证明，仅指出一种该构造的变体，其顶点数更多，但比值$\frac{a(G)}{n}=\frac{37}{76}$略小。
英文摘要
For a graph $G$, denote by $a(G)$ the number of vertices in the largest induced forest in $G$. The Albertson-Berman conjecture, which had been open since 1979, states that $a(G) \geq \frac{n}{2}$ for every simple planar graph $G$ on $n$ vertices. Although the Albertson-Berman conjecture was recently resolved in the negative by constructing a counterexample with the help of AI, we independently found a counterexample to the Albertson-Berman conjecture without AI and present it in this article. Our counterexample is on $39$ vertices with $a(G)=19$. Finally, we indicate, without a full proof, a variant of this construction with a larger number of vertices, but with a slightly smaller ratio $\frac{a(G)}{n}=\frac{37}{76}$.
