Colored Quadrangulations with Steiner Points

Let P be a k-colored set of n points in general position on the plane, where k ≥ 2. A k-colored quadrangulation of P is a maximal straight-edge plane graph with vertex set P satisfying the property that every interior face is a properly colored quadrilateral, i.e., no edge connects vertices of the same color. It is easy to check that in general not every set of points admits a k-colored quadrangulation, and hence the use of extra points, for which we can choose the color among the k available colors, is required in order to obtain one. The extra points are known in the literature as Steiner points. In this paper, we show that if P satisfies some condition for the colors of the points in Conv(P), then a k-colored quadrangulation of P can always be constructed using less than (16k−2)n+7k−2 39k−6 Steiner points. Our upper bound improves the previously known upper bound for k = 3, and represents the first bounds for k ≥ 4.