1-page and 2-page drawings with bounded number of crossings per edge

A drawing of a graph such that the vertices are drawn as points along a line and each edge is a circular arc in one of the two half-planes defined by this line is called a 2-page drawing. If all edges are in the same half-plane, the drawing is called a 1-page drawing. We want to compute 1-page and 2-page drawings of planar graphs such that the number of crossings per edge does not depend on the number of vertices. We show that for any constant k, there exist planar graphs that require more than k crossings per edge in both 1-page and 2-page drawings. We then prove that if the vertex degree is bounded by Δ, every planar 3-tree has a 2-page drawing with a number of crossings per edge that only depends on Δ. Finally, we show a similar result for 1-page drawings of partial 2-trees.