A counterexample to the Albertson-Berman conjecture about induced forests in planar graphs

Abstract

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 $a(G)\ge \frac{n}{2}$ 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 $\frac{a(G)}{n}=\frac{37}{76}$.

arXiv comment15 pages, 5 figures