Exact value of the optimal induced-forest ratio for planar graphs

Determine the exact value of the optimal constant c = inf a(G)/|V(G)| over all planar graphs, improving the currently known bounds 2/5 ≤ c ≤ 15/31 where possible.

Background

The paper defines c as the infimum, over planar graphs G, of the ratio between the maximum order a(G) of an induced forest and the number of vertices |V(G)|. Borodin’s acyclic 5-coloring theorem gives the universal lower bound c ≥ 2/5, while the constructed family of simple planar graphs M_k has a(M_k)/|V(M_k)| = 15/31, yielding c ≤ 15/31.

The authors explicitly state that determining c, or improving either endpoint of this interval, remains unresolved. The question concerns the sharp asymptotic and finite extremal behavior of induced forests in planar graphs.

References

Thus

\frac25\le c\le\frac{15}{31}.

Determining $c$, or improving either side of this interval, remains open.

We close with three specific open questions.

\begin{enumerate} \item What is the exact value of~$c$? The current range $2/5\le c\le 15/31$ is wide, and there is room to improve both the lower and upper bounds.

— A 15/31 Counterexample Family to the Albertson-Berman Conjecture  (2608.17350 - Jung, 18 Aug 2026) in Section 7, Discussion and open problems; first enumerated question

\item What is the smallest counterexample to the conjecture? The seed~$T$ has $31$ vertices, but this is the minimum within our construction; smaller counterexamples from other constructions may well exist.

— A 15/31 Counterexample Family to the Albertson-Berman Conjecture  (2608.17350 - Jung, 18 Aug 2026) in Section 7, Discussion and open problems; third enumerated question

We conclude this paper by listing several natural open problems that remain. The results discussed in this paper imply that \frac{2}{5} \leq c_{\mathcal{P} \leq \frac{25}{52}. Improve these bounds.

— Counterexamples to the Albertson-Berman conjecture: minimum order, connectivity and an improved ratio bound  (2608.23260 - Batenburg et al., 24 Aug 2026) in Section 5, “Open problems”