Graph Treewidth and Geometric Thickness Parameters

Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal dk/2e. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity. ? This research was initiated while both authors were in the School of Computer Science at McGill University, Montréal, Canada. A preliminary version of this paper was published in the Proceedings of the 13th International Symposium on Graph Drawing (GD ’05), Lecture Notes in Computer Science 3843:129–140, Springer, 2006. ?? Partially supported by NSERC, Centre de Recherches Mathématiques (CRM), and Institut des Sciences Mathématiques (ISM). ??? Supported by a Marie Curie Fellowship of the European Community under contract 023865, and by the projects MCYT-FEDER BFM2003-00368 and Gen.