Improve the upper bound for planar graph track number

Improve the upper bound of $225$ on the track number of planar graphs.

Background

The paper’s main product-structure framework attains the current best-known upper bound of $225$ for planar graphs. The authors explicitly identify improving this bound as a separate open problem, since any improvement could lead to stronger bounds for the broader graph families treated through product decompositions.

The problem remains unresolved in the paper: the authors state that their efforts did not produce an improvement, emphasizing the difficulty and importance of reducing the planar-graph track-number bound.

References

As a second open problem, we highlight the challenge of improving the upper bound of $225$ on the track number of planar graphs, which our results match. Progress in this direction could provide valuable insights toward further improvements to the bounds established in this work. Despite our efforts towards this goal, however, an improvement remains elusive, highlighting both the difficulty of the problem and its fundamental importance in this area of research.

— Product Structure Meets Track Layouts  (2608.27096 - Bekos et al., 27 Aug 2026) in Section 5, Conclusions