Colored Quadrangulations with Steiner Points

Abstract

Let P ⊂ ℝ² be a k-colored set of n points in general position, where k ≥ 2. A k-colored quadrangulation on P is a properly colored straight-edge plane graph G with vertex set P such that the boundary of the unbounded face of G coincides with CH(P) and that each bounded face of G is quadrilateral, where CH(P) stands for the boundary of the convex hull of P. It is easily checked that in general not every k-colored P admits a k-colored quadrangulation, and hence we need the use of Steiner points, that is, auxiliary points which can be put in any position of the interior of the convex hull of P and can have any color among the k colors. In this paper, we show that if P satisfies some condition for colors of the points in the convex hull, then a k-colored quadrangulation of P can always be constructed using less than $\frac{(16 k-2) n+7 k-2}{39 k-6}$ Steiner points. Our upper bound improves the known upper bound for k = 3, and gives the first bounds for k ≥ 4.