Improved Bounds for Track Numbers of Planar Graphs

TL;DR

This paper advances the understanding of track numbers in planar graphs by establishing new upper bounds for various subclasses and providing improved lower bounds, enhancing the theoretical framework of graph layout complexity.

Contribution

It introduces tighter bounds on the track number for planar graphs, 3-trees, outerplanar graphs, and other specialized graph classes, with new lower bounds for outerplanar graphs.

Findings

Planar graphs have a track number at most 225.

Planar 3-trees have a track number at most 25.

Outerplanar graphs can have a track number of at least 5.

Abstract

A track layout of a graph consists of a vertex coloring and a total order of each color class, such that no two edges cross between any two color classes. The track number of a graph is the minimum number of colors required by a track layout of the graph. This paper improves lower and upper bounds on the track number of several families of planar graphs. We prove that every planar graph has track number at most $225$ and every planar $3$-tree has track number at most $25$. Then we show that there exist outerplanar graphs whose track number is $5$, which leads to the best known lower bound of $8$ for planar graphs. Finally, we investigate leveled planar graphs and tighten bounds on the track number of weakly leveled graphs, Halin graphs, and X-trees.