Track Layouts of Graphs† - Discrete Mathematics & Theoretical ...

Discrete Mathematics and Theoretical Computer Science 6, 2004, 497–522

Track Layouts of Graphs† Vida Dujmovi´c1,2 and Attila P´or3 and David R. Wood1,2 1 2 3

School of Computer Science, McGill University, Montr´eal, Canada. ({vida,wood}@cs.mcgill.ca) School of Computer Science, Carleton University, Ottawa, Canada. Department of Mathematics, Case Western University, Cleveland, USA. ([email protected])

received November 4, 2003, revised September 7, 2004, accepted December 2, 2004.

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.

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, acyclic chromatic number Mathematics Subject Classification: 05C62 (graph representations), 05C15 (coloring of graphs and hypergraphs)

† The research of V. Dujmovi´ c was supported by FCAR and NSERC. The research of D. Wood was partially supported by NSERC. The research of A. Por and D. Wood was partially completed at the Department of Applied Mathematics and the Institute for Theoretical Computer Science, Charles University, Prague, Czech Republic. Supported by project LN00A056 of the Ministry of Education of the Czech Republic, and by the European Union Research Training Network COMBSTRU (Combinatorial Structure of Intractable Problems).

c 2004 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France 1365–8050

498

1

Vida Dujmovi´c and Attila P´or and David R. Wood

Introduction

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. The purpose of this paper is to develop the beginnings of a theory of track layouts. Our focus is on methods for the manipulation of track layouts, and the relationship between track layouts and other models of graph layout. We consider undirected, finite, and simple graphs G with vertex set V (G) and edge set E(G). The number of vertices and edges of G are respectively denoted by n = |V (G)| and m = |E(G)|. A vertex |I|-colouring of a graph G is a partition {Vi : i ∈ I} of V (G) such that for every edge vw ∈ E(G), if v ∈ Vi and w ∈ V j then i 6= j. The elements of I are colours, and each set Vi is a colour class. The chromatic number of G, denoted by χ(G), is the minimum number k such that G has a vertex k-colouring. Consider a vertex I-colouring {Vi : i ∈ I} of a graph G. Suppose that