Journal of Graph Algorithms and Applications
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Improved Bounds for Track Numbers of Planar Graphs
Vol. 24, no. 3, pp. 323-341, 2020. Regular paper.
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.
Submitted: November 2019.
Reviewed: March 2020.
Revised: April 2020.
Accepted: June 2020.
Final: June 2020.
Published: July 2020.
Communicated by Giuseppe Liotta