Establish a linear lower bound on the independence number of biplanar graphs

Determine whether there exists a constant c>1/12 such that every biplanar graph on n vertices has independence number at least cn.

Background

This is identified as the second of two smaller questions that remain open and is attributed to Kral'. The paper proves only that every biplanar graph on at least 19 vertices has an independent set of size at least 3. That bound improves on the immediate degeneracy consequence for some small orders but does not establish a linear lower bound with a constant greater than 1/12.

References

The second is the question of Kral'~\citep[Problem~4]{barbados2018} whether every biplanar graph on $n$ vertices has $\alpha\ge cn$ for some constant $c>1/12$.

— Biplanar graphs with independence number two are 9-colorable  (2609.28102 - Szeider, 23 Sep 2026) in Section 6, Concluding remarks