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 θ(G). By restricting the edges to be straight, we obtain the geometric thickness θ̄(G). By further restricting the vertices to be in convex position, we obtain the book thickness bt(G). This paper studies the relationship between these parameters and the treewidth of G. Let θ(Τk) / θ̄(Τ k) / bt(Τk) denote the maximum thickness / geometric thickness / book thickness of a graph with treewidth at most k. We prove that: - θ(Τk) = θ̄(Τk) = [k/2], and - bt(Τk) = k for k ≤ 2, and bt(Τk) = k + 1 for k ≥ 3. The first result says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. The second result disproves the conjecture of Ganley and Heath [Discrete Appl, Math. 2001] that bt(Τk) = k for all k. Analogous results are proved for outerthickness, arboricity, and star-arboricity.
Dujmović, V, & Wood, D. (2006). Graph treewidth and geometric thickness parameters.