Improved Bounds for Track Numbers of Planar Graphs
share
A track layout of a graph consists of a vertex coloring and of a total order of each color class, such that the edges between each pair of colors form a non-crossing set. 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 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.
READ FULL TEXT