Discrete Mathematics & Theoretical Computer Science, Vol 6, No 2 (2004)

Font Size:  Small  Medium  Large
DMTCS vol 6 no 2 (2004), pp. 497-522

Discrete Mathematics & Theoretical Computer Science


Volume 6 n° 2 (2004), pp. 497-522

author:Vida Dujmović and Attila Pór and David R. Wood
title:Track Layouts of Graphs
keywords:graph layout, graph drawing, track layout, stack layout, queue layout, book embedding, track-number, queue-number, stack-number, page-number, book-thickness, geometric thickness, three-dimensional graph drawing
abstract:A (k,t)-track layout of a graph G consists of a (proper) vertex t-colouring of G, a total order of each vertex colour class, and a (non-proper) edge k-colouring such that between each pair of colour classes no two monochromatic edges cross. This structure has recently arisen in the study of three-dimensional graph drawings. This paper presents the beginnings of a theory of track layouts. First we determine the maximum number of edges in a (k,t)-track layout, and show how to colour the edges given fixed linear orderings of the vertex colour classes. We then describe methods for the manipulation of track layouts. For example, we show how to decrease the number of edge colours in a track layout at the expense of increasing the number of tracks, and vice versa. We then study the relationship between track layouts and other models of graph layout, namely stack and queue layouts, and geometric thickness. One of our principle results is that the queue-number and track-number of a graph are tied, in the sense that one is bounded by a function of the other. As corollaries we prove that acyclic chromatic number is bounded by both queue-number and stack-number. Finally we consider track layouts of planar graphs. While it is an open problem whether planar graphs have bounded track-number, we prove bounds on the track-number of outerplanar graphs, and give the best known lower bound on the track-number of planar graphs.

If your browser does not display the abstract correctly (because of the different mathematical symbols) you can look it up in the PostScript or PDF files.

reference: Vida Dujmović and Attila Pór and David R. Wood (2004), Track Layouts of Graphs, Discrete Mathematics and Theoretical Computer Science 6, pp. 497-522
bibtex:For a corresponding BibTeX entry, please consider our BibTeX-file.
ps.gz-source:dm060221.ps.gz (224 K)
ps-source:dm060221.ps (849 K)
pdf-source:dm060221.pdf (266 K)

The first source gives you the `gzipped' PostScript, the second the plain PostScript and the third the format for the Adobe accrobat reader. Depending on the installation of your web browser, at least one of these should (after some amount of time) pop up a window for you that shows the full article. If this is not the case, you should contact your system administrator to install your browser correctly.

Due to limitations of your local software, the two formats may show up differently on your screen. If eg you use xpdf to visualize pdf, some of the graphics in the file may not come across. On the other hand, pdf has a capacity of giving links to sections, bibliography and external references that will not appear with PostScript.

Automatically produced on Sun Dec 12 11:12:13 CET 2004 by falk