Planar-graph scramble number square-root bound

Prove or disprove that the scramble number of every planar graph on n vertices is bounded by O(√n).

Background

The paper studies the scramble number as a graph invariant related to treewidth and gonality. Prior work established an O(√n) bound for planar graphs of bounded maximum degree, while the paper proves an explicit O(√n) bound for outerplanar graphs and certain nearly outerplanar graphs. The authors identify the extension to all planar graphs as an unresolved conjecture; their outerplanar result shows that any counterexample must be far from outerplanar.

References

In the authors proved that planar graphs of bounded degree have scramble number at most $O(\sqrt{n})$ and they also conjecture that for all planar graphs the scramble number is bounded by $O(\sqrt{n})$.

— The scramble number of outerplanar graphs  (2609.03755 - Laboy, 3 Sep 2026) in Section 1, Introduction