Abstract
A \emph(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.\
Highlights
In its simplest form, a track layout of a graph consists of a vertex colouring and a total order on each colour class, such that there is no pair of crossing edges between any two colour classes
As corollaries we prove that acyclic chromatic number is bounded by both stack-number and queue-number
A graph G has a 1-queue layout if and only if G has a track layout {Vi : 1 ≤ i ≤ t} with maximum span two, such that for every edge vw ∈ E(G) that has span two (v ∈ Vi, w ∈ Vi+2), w is the first vertex in Vi+2, and there is no edge xy ∈ E(G) with v
Summary
In its simplest form, a track layout of a graph consists of a vertex colouring and a total order on each colour class, such that there is no pair of crossing edges between any two colour classes. An X-crossing in a track assignment consists of two edges vw and xy such that v, x ∈ Vi, w, y ∈ Vj, v
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