combinatorics / Graph theory
Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.
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The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.
Research memory
Albertson and Berman conjectured that for every simple planar graph $G$ on $n$ vertices, the largest vertex set inducing a forest has size at least $n/2$. The standing lower bound since the same year has been Borodin's $2n/5$, from his acyclic five-colour theorem. False: there is an explicit $31$-vertex simple $3$-connected maximal planar graph $T$ whose largest induced forest has exactly $15$ vertices, and an infinite family $M_k$ on $31k$ vertices with induced-forest number exactly $15k$, giving the ratio $15/31 < 1/2$ even for triangulations of minimum degree five.
The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.
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