Quadrangulations on 3-Colored Point Sets with Steiner Points and Their Winding Numbers

Abstract

Let P be a point set on the plane, and consider whether P is quadrangulatable, that is, whether there exists a 2-connected plane graph G with each edge a straight segment such that V(G) = P, that the outer cycle of G coincides with the convex hull Conv(P) of P, and that each finite face of G is quadrilateral. It is easy to see that it is possible if and only if an even number of points of P lie on Conv(P). Hence we give a k-coloring to P, and consider the same problem, avoiding edges joining two vertices of P with the same color. In this case, we always assume that the number of points of P lying on Conv(P) is even and that any two consecutive points on Conv(P) have distinct colors. However, for every k ≥ 2, there is a k-colored non-quadrangulatable point set P. So we introduce Steiner points, which can be put in any position of the interior of Conv(P) and each of which may be colored by any of the k colors. When k = 2, Alvarez et al. proved that if a point set P on the plane consists of ${\frac{n}{2}}$ red and ${\frac{n}{2}}$ blue points in general position, then adding Steiner points Q with ${|Q| \leq \lfloor \frac{n-2}{6} \rfloor + \lfloor \frac{n}{4} \rfloor +1}$ , P ∪ Q is quadrangulatable, but there exists a non-quadrangulatable 3-colored point set for which no matter how many Steiner points are added. In this paper, we define the winding number for a 3-colored point set P, and prove that a 3-colored point set P in general position with a finite set Q of Steiner points added is quadrangulatable if and only if the winding number of P is zero. When P ∪ Q is quadrangulatable, we prove ${|Q| \leq \frac{7n+34m-48}{18}}$ , where |P| = n and the number of points of P in Conv(P) is 2m.