Improve track-number bounds for k-trees and planar 3-trees

Improve the upper bound of $(k+1)(2^{k+1}-2)^k$ on the track number of $k$-trees, including, in particular, the current upper bound of $25$ on the track number of planar $3$-trees.

Background

The paper identifies the track number of kk-trees as a central unresolved issue because the bound established by Wiechert, (k+1)(2k+1−2)k(k+1)(2^{k+1}-2)^k, is used to derive corresponding bounds for several graph families with product structures. Improving this bound would therefore immediately improve the track-number bounds for most of the graph classes summarized in the paper.

The planar $3$-tree case is singled out as especially important: multiple graph classes can be represented using strong products involving a planar $3$-tree, and improving Pupyrev’s upper bound of $25$ for planar $3$-trees would consequently yield stronger results for those classes.

References

A central open problem is to improve the upper bound of $(k+1)(2{k+1}-2)k$ on the track number of $k$-trees by Wiechert. Any progress in this direction would immediately translate into corresponding improved bounds for the track number of most graph families listed in \cref{tbl:results}. In particular, the case $k = 3$ under the additional assumption of planarity is of special interest, as several graph classes can be expressed in terms of the strong product of simple graphs and a planar $3$-tree. In this regard, improving the current upper bound of $25$ on the track number of planar $3$-trees due to Pupyrev would have significant implications for these classes as well.

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