# Track Layouts of Graphs

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1 School of Computer Science Ottawa 2 SOCS - School of Computer Science Quebec 3 Department of Mathematics, Applied Mathematics and Statistics Cleveland 4 KAM - Department of Applied Mathematics Prague

Abstract : A \emphk,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.\

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

Author: ** Vida Dujmović - Attila Pór - David Wood - **

Source: https://hal.archives-ouvertes.fr/